POSSIBLE STABILITY OF POLYMORPHISM IN A MULTIALLELE INCOMPATIBILITY SYSTEM IN THE MOUSE

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INCOMPATIBILITY SYSTEM IN THE MOUSE1

PETER HULL

Department of Genetics, The University, Liverpool 3 , England

Received July 8, 1966

H E conditions affecting the existence and stability of a genetic polymorphism in the mouse, produced by incompatibility between mother and offspring of the same genotype have been examined. In the case considered (HULL 1964) it was found that the probability of survival of an at/at offspring produced by a n a t / a f mother, and the probability of survival of an at/+ offspring produced by a n a'/+ mother were both reduced. Although the observed selective parameters were such that no stable polymorphism could exist, it was possible to show that stable equilibria could exist in a system of this sort, and to obtain the general conditions for stability.

It has been shown that immunological dissimilarity of mother and foetus fa- vours foetal development (reflected by placental growth) in the mouse (BIL-

LINGTON 1964). Furthermore, in the instance of partial incompatibility associated

with the +/at system it was noted that it was possible that the observed incompatibility might be due either to the effect of the two genes f and at themselves, or to a gene or genes, perhaps histocompatibility genes, sufficiently closely linked to them to remain associated through several meiotic divisions. I t is notable that several histocompatibility loci of the mouse are characterised by having a number of alleles: see for example SNELL (1958). It therefore appeared of interest to see whether an autoincompatibility system of the type already examined was capable of maintaining a number of alleles in a stable equilibrium, and if so, under what conditions, or whether the maximum number of alleles which could be retained in such a system was two.

THE MODEL

Considering the alleles A,,A,

. . .

A , at a single locus, the conditions where all n alleles would be kept segregating can be investigated. The effect is assumed, as in the two allele case, to be one where the viability of offspring of like genotype to their mother is reduced, and where selective elimination takes place in the uterus at a sufficiently early time in a multiparous animal like the mouse when a n excess of embryos is available so that total litter size is not affected by the selection.

If we let (1 - s , ~ ) be the probability of survival of a n A,Aj offspring from an

A , A , mother, and p," be the initial frequency of allele A , , where x,pz = 1, then ' Pal t of thir W O I k was done during a D S I R /N A T 0 Fellowship

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1050 _{P. HULL}

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1051

it is possible to write down the frequency, after one generation of random mating and selection of the [ 1/2n ( n

+

1 ) ] types of mating and the 1/2n (n' $1 ) types of off spring, assuming that the population is in Hardy-Weinberg equilibrium. TO

simplify the succeeding algebra it is convenient to write tij for s i j / ( 2 - sii). The frequencies of mating types and offspring are listed in Table 1.

The change in frequency of allele A i in one generation will be

n

Api =z

1/2

f i j f f i i - pio,

;,a

where f i i is the frequency of

AiAi

individuals and f i i is the frequency of AiAj

individuals.

Api can be regarded as approximately d p i / d t , the rate of change of pi with time: we have n - 1 independent equations of this type which can be written

t -thh tji-thi t' ' - t h i

Aph

5

ph3pi

[

%]

f &hpi3

[

7 +

5 5

]

phpi'pj

[e]

.

i = 1 i = 1 i = l j = 1

i # h i # h i # h , j # h i#i

MU LT I AL LEL E EQ U ILIBR I UM I N INDIVIDUAL CASES

If we consider first the case where selective elimination of all homozygotes is equal, as is that of all heterozygotes, i.e.

we have, from the Api at equilibrium

Y

giving directly, as expected

$ 1 = $2 =

. . .

$6

.

. .

=

pn.

The conditions necessary for the stability of this equilibrium can be obtained by examining the characteristic roots associated with the matrix

B

whose elements are b,,, where b,, =

-

for T- = 1

.

n - 1, c = 1

.

n - 1, at equilibrium. The equilibrium will be stable if the largest characteristic root is negative. Under these circumstances

a P C

t0-te

.

Aph=$ ph3pi

[y

]

+

5

phpi3

[

7 1

2 = 1 i = 1

i#h i # h So the diagonal elements evaluated at equilibrium are _-

aapi te-to

--

-

pi3

[

-1

E

(3pizpj-pi3).

api j = 1

But remembering all $i are equal, all diagonal elements are thus

Nondiagonal elements

( i

# j ) are:

-

aApi - - pi

[

t+] ( p 1 - p ;

+

3p;

-

3 p 3 = 0.

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1052 P. HULL

Thus all roots will be negative when to

>

t,. That is, a stable equilibrium involving all alleles is possible and will exist if the selective elimination of homozygotes is greater than that of heterozygotes.

In the case where selective elimination differs among homozygotes, but is the same for all heterozygotes, i.e.

tll # t 2 2 # . .

.

# t,,, but t i j = t, as before,

it is now found from the Api at equilibrium, i.e.

pi

Api = 0 =

[

_p;tij- p f t i - pj3te -I- pi?,

]

So the diagonal elements evaluated at equilibrium are

1

aApi - t - tic ll.

api j = 1

-

- P z ,

[ +]

2pj == p t

[

t e - tii

whereas nondiagonal elements are, as before

(i

# j )

-= anpi

[+]

t -tii

(

p 3 i - p ; + 3 p ; - 3 p ; )

= o

.

api

Thus all roots will be negative, giving stability at equilibrium when t,,

>

t , for

i

= 1

.

n. That is, a stable equilibrium involving all alleles is possible and will exist if selective elimination of all different types of homozygotes is greater than that of heterozygotes.

When selective elimination differs among the heterozygotes it was not found possible to use this approach as no simple relation among the $,'s could be found. It is however still possible to obtain necessary conditions for the stability of equilibrium, assuming that there is at most one point of equilibrium (i.e. where all p, f 0) for this will be the case for those conditions which ensure that all the points with p z = 1 are simultaneously points of instability. We find that for p , = I the diagonal elements of B are:

t,, - t , j , (j=1 .

. .

n,

j#i)

aAPj -

aP3

--

whereas all other elements except those of the ith row are zero. For instability at p 2 = 1 we thus have the n-1 requirements that

t,,

>

t,,, (i=1

.

n,

i#i).

If there is only one point of stability for a given set of t,, with all alleles segregating, we then have the n(n-1) simultaneous requirements

t,,

>

t,,, (i=l

. . .

n; j=1

. .

. n, jZi).

That is, a necessary condition for a stable equilibrium to exist involving all alleles is that the selective elimination of homozygotes of each allele must be greater than the selective elimination for all heterozygotes of that allele.

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S U M M A R Y

It is possible to show that a stable polymorphism involving more than two alleles could exist under certain circumstances with maternal-foetal incompatibility. Necessary conditions for the existence of this polymorphism are given for several cases.

LITERATURE C I T E D

BILLINGTON, W. D., 1964 HULL, P., 1964

Influence of immunological dissimilarity of mother and foetus on the

Partial incompatibility not affecting total litter size in the mouse. Genetics 50:

Histocompatibility genes of the mouse. 11. Production and analysis of isogenic size of placenta in mice. Nature 202: 317-318.

563-570. SNELL, G. D., 1958