Mathematical model for the study of fertility and viability in inbreeding populations

Bulletin of Mathematical Biology Vol. 44, No. 4, pp. 491-500, 1982. Printed in Great Britain

0092-8240/82/040491- 10503.0010 Pergamon Press Ltd © 1982 Society for Mathematical Biology

MATHEMATICAL MODEL FOR THE STUDY OF FERTILITY AND VIABILITY IN INBREEDING POPULATIONS •

A. R. BOUFFETTE and J. BOUFFETTE D6partement de Math6matiques, Universit6 Claude-Bernard, L y o n 1, France

Granted that a single or complex gene is responsible for inbreeding depression, theoretical expressions for fertility and viability are obtained in different diploid populations: brother-sister, half-brother-sister, cousins and double-cousins. The conclusions of the study of viability variations according to the coefficient of parentage are proved by the results of experiments and lead to a new view of genetic load.

Introduction. Making inbreedings, essentially of the brother-sister type, on Drosophila melanogaster, Biemont (1978 1980) revealed the existence of an inbreeding sensitivity gene, Is, which is capable of regulating the embryonic then larvo-pupal development in a different way according to the parental couple from which the fecund egg is issued. A consanguineous population is then characterised by the effects of the Is gene, that is to say, by the average fertility and average viability of this population. In this study, our aim is to calculate, for various types of inbreedings, the fertility and viability of a couple taken at random from the studied consanguineous population, in order to check the model given by Biemont (1980). The analysis enables one to foresee the effect of various interbreedings other than brother-sister ones, which leads to a new interpretation of the genetic load concept. The genotypes of diploid zygotes considered in this study are written AA, Aa, aa, which implies that the gametes which they are issued from bear either the A gene or the a gene. We will distinguish two stages in the development of a fecund egg: Stage 1, the embryonic one, during which we admit that only a fecund egg issued from a couple of the (aa x aa) type has a probability rn not to reach the larval state. Stage 2, the larvo-pupal one, along which any larve (aa), whatever the parental couple, has the probability k of dying before becoming an adult. We will represent the influence of inbreeding by giving to rn and k the 491

492

A . R . B O U F F E T T E A N D J. B O U F F E T T E

value 0 if the f e c u n d egg is issued f r o m parents with no c o m m o n ancestor, and strictly positive values if the f e c u n d egg is issued f r o m parents with at least one c o m m o n ancestor. The probability for a f e c u n d egg, issued f r o m the inbreeding of two individuals I and J, to give rise to a larva is written F(I, J), and is called fertility of the couple (I, J). In the same way, the probability for a f e c u n d egg to give rise to an adult is d e n o t e d by V(L J), and is t e r m e d viability

of the couple (L J). We assume that, if I and J have no c o m m o n ancestor, then

F(L J) = 1 and V(L J) = 1.

(1)

On the contrary, w h e n I and J have one c o m m o n ancestor at least, we can make out six possibilities c o r r e s p o n d i n g to the following situations: $1: a a x aa $2: Aa x aa $3: Aa x Aa

$4: A A x aa $5: A A x Aa $6: A A x AA.

Writing respectively Fj and Vj, the fertility and the viability of the couple (/, J) when it is in the situation Sj (j = 1, 2 . . . . . 6), we get immediately the expressions of the fertility F(I, J) and of the viability V(I, J): 6

F(I, J) = Y~ Fj. P(S~),

(2)

j=l 6

V(I, J) = ~ Vj. P(Sj),

(3)

j=l

where P(Sj) represents the probability for the couple (/, J) to be in the situation S i. We have already noted that only a f e c u n d egg issued f r o m a couple in the situation $1 can die during the e m b r y o n i c stage, so that F1 = 1 - m

(4)

and F j = 1 for j = 2 , 3, 4, 5, 6;

(5)

F(I, J) = 1 - m • P(S1).

(6)

that is to say:

MATHEMATICAL MODEL FOR THE STUDY OF FERTILITY

493

On the contrary, any larva of the (aa) genotype cannot reach adult age, therefore

V1 (1 ----

-

m)(1

-

k),

1

V3 = 1-~k, (7)

V2=Vs=V6=l;

V2-- 1-½k, that is to say" V(LJ) = I - m ( l - k ) .

P(S,)-k[P(S,)+½P(S2)+~P(Sa)].

(8)

We note that the expression P(S1)+ ½P(S2)+1p(s3) can be interpreted as being the probability, termed P*(aa), for a fecund egg taken at random among the fecund eggs of the couple (L J) to come from the union of two gametes bearing the a gene. Consequently, we have

F ( L J) - V(L J) = k[P*(aa) - raP(SOl.

(9)

General Expressions o f Fertility and Viability. Let us take two individuals I and J. If we set l(I, J ) - - 0 , if I and J have no common ancestor, and l(I, J) = 1 if I and J have one common ancestor at least, then then F ( L J) = 1 - m • l(I, J ) . P(S,),

(10)

V(L J) = 1 - I(L J){m(l - k)P( S,) + k[P(S,) + ½P(S2) +-~P(S3)]}.

(11)

Use o f the Coe~icients o f Restricted Identity. If Ar is the coefficient of restricted identity corresponding to the rth situation of identity introduced by Gillois (1965) and Jacquard (1970), and if at(q) is the probability for the four genes, supposed to be in the situation of identity r, to be altogether a genes, then P ( S , ) is simply written 9 P(Sl)-=

Z Ar" °G(q)"

(12)

r=l

On the other hand, we have P*(aa) = ~uq + (1 - q~ij)q2,

(13)

494

A . R . B O U F F E T T E A N D J. B O U F F E T T E

an expression in w h i c h q is the f r e q u e n c y of the a gene in the generation of the individuals I and J, and ~ , their coefficient of parentage. Consequently: 9

F(L J ) = 1 - m

~, A . a~(q),

(14)

r=l 9

V(L J) = 1 - (1 - k)m ~, Ar . a~(q)- k[q~u " q + ( 1 - ~u)" q2].

(15)

r=l

As at(q) are p o w e r s of q, F(I, J) decreases f r o m 1 to (1 - m) w h e n q varies from 0 to 1. Likewise, V(L J) is also a decreasing f u n c t i o n of q and varies f r o m 1 to ( 1 - m ) ( 1 - k). T h e n , for a couple (/, J), we always have 1 - rn <~ F ( L J) ~ 1

(16)

(1 - m ) ( 1 - k) <~ V(I, J) <~ 1.

(17)

Fertility, Viability and Inbreeding.

S u p p o s e that the individuals of the original situation Go are not c o n s a n g u i n e o u s , that is to say they have no c o m m o n ancestor. In these conditions, it is possible to m a k e a first study u p o n the fertility and viability of the following c o n s a n g u i n e o u s generation GI.

Fertility and inbreeding. T h e fertility of a couple (/, J) is i m m e d i a t e l y given by the f o r m u l a (1) and the k n o w n values of Ar and at(q). In the following specific cases, these values are s u m m e d up in Table I. The fertility of a b r o t h e r - s i s t e r couple is m

FFs(L J) = 1 - -~- q2(1 + q)2,

(18)

whereas the fertility of a half-brother-sister couple is m

F1/zvs(L J) = 1 --~- q3(1 + q).

(19)

Since m

F,/zFs(I, J) - FFs(I, J) = q q2(1 - q2),

(20)

MATHEMATICAL MODEL FOR THE STUDY OF FERTILITY

495

TABLE I Coefficients of Restricted Identity At: Specific Values for Brother-Sister (FS), Half-Brother-Sister (1/2FS), Cousins (C), Double-Cousins (2C) Inbreedings

c,o

G j B O G *

.

IIII

.

~

:

•

II

•

,

,

~

II

_

.

•

II

Ar

AI

A2

A3

A4

A5

A6

F.S

0

0

0

0

0

0

~FS C

0 0

0 0

0 0

0 0

0 0

0 0

0 0

2C

0

0

0

0

0

0

~

Otr(q)

q

02

q2

q3

q2

q3

,

,

I

,

,

,

A7

A8

A9

~IJ

~

~

~

I

1

~1 3 ~ 9

1 1 r~ 1

q2 1

1

~3

~1 ~I 6

s~

q4

~

we have FFs(I, J) <- Fl/2Fs(I, J).

(21)

Hence, in generation G1 the couples of individuals have fertilities which can be arranged in the reverse order of their respective coefficient of parentage. Likewise, in generation G2 we obtain a similar result when comparing the fertility of a cousins couple with the fertility of a double-cousins couple: m

Fc(I, J) = 1 --~- q3(1 + 3q),

F2~(L J) = 1 - ~

m

q2(1 + 3q) 2,

(22)

(23)

so that

F2c(t, J) <- Fc(I, J).

(24)

We can also consider, in generation G2, a brother-sister couple and a half-brother-sister couple, couples composed of individuals issued from non-consanguineous parents of generation GI. Then FFS(I, J) <- F2c(I, J)<~ F1/2Fs(I, J ) ~ Fc(I, J).

(25)

496

A . R . B O U F F E T T E A N D J. B O U F F E T T E

Inbreeding and viability.

In order to calculate the viability of a couple (/, J) it is possible to use f o r m u l a (2) or the following equivalent expression: expression: 9

m(1

V(L J) = 1 - kq a -

- k) ~.

At.

ar(q)

r=l

-

kq(1

-

q)[A, + ½(A3+ A5 + A7) + ~As].

(26)

Hence, the viability of a b r o t h e r - s i s t e r couple is given by

VFS(I, J) = 1 - ¼m(1 - k)qa(1 + q)2 _ ~kq(1 + 3q),

(27)

that of a half-brother-sister couple by

V1/2FS(I,J) = 1 -½m(1

k)q3(l + q) - - ~ k q ( 1 + 7q),

(28)

Vc(L J) = 1 - ~ 1m ( 1 - k)q3(1 + 3q) - ~ k q ( 1 + 15q),

(29)

-

that of a cousins couple by

and that of a double-cousins couple by

V2c(L J) = 1 -~6m(1 - k)q2(1 + 3q) 2 - ~kq(1 + 7q).

T A B L E II E x p e r i m e n t a l Values of Viability T = 0

C ~p = ~

½FS l

~o = ~

FS l

~o = ~

90.2

83.2

--

80.5

69.7

59.7

58.8

55.2

84.3

--

76.0

63.5

86.9

--

77.5

73.8

Authors S t o n e et al. (1963) Mettler and Gregg (1969) MalogolowkinC o h e n et al. (1964) Dobzhansky

et al. 86.4

--

82.9

82.6

76.9

71.7

--

64.8

(1963) Toroja (1964) S t o n e et al. (1963)

(30)

MATHEMATICAL MODEL FOR THE STUDY OF FERTILITY

497

C o n s e q u e n t l y , we are led to the same ordering relations (and to the same r e m a r k s 1 and 2) as for fertility, n a m e l y VFs(I, J) <~ VI/2Fs(L J)

(31)

V2c(I, J) <~ Vc(I, J)

(32)

VFs(L J) <~ V2c(I, J) ~ G/2Fs(/, J) <~ Vc(L J).

(33)

We note the following important points: Relations (25) and (33) are still true b e t w e e n the fertilities and viabilities of the b r o t h e r - s i s t e r or half-brother-sister couples of generation GI, and those of the cousins and double-cousins of generation G2, in the condition that the f r e q u e n c y of the a gene is the same in both generations. Neither fertility, nor viability are functions of the parentage coefficient. As a matter of fact, the coefficient of parentage of a half-brother-sister couple is equal to one-eighth of that of a double-cousins couple, whereas:

F~c(I, J) <- F,/~,~s(I, J),

(34)

V2c(L J) <- V,/:~s(I, J).

(35)

These last results would tend to prove that fertility and viability are good clues to r e p r e s e n t the effects of inbreeding, e v e n better than the coefficient of parentage, for t h e y also take a certain cytoplasmic heredity into account. The c o m p a r i s o n b e t w e e n the viabilities of couples, taken f r o m a similar generation, allows one to write

VFs(I, J) = ¼m(1 - k)q2(1 - q2)+ ~kq(1 - q),

(36)

V c ( L J) - V,/2Fs(L J) = ~m(1 - k)q3(1 - q ) + ~ k q ( 1 - q),

(37)

V,/2Fs(I, J ) -

V2c(I, J) - VFs(I, J) = ~6m(1 - k)q2(1 - q)(5q + 3) + ~kq(1 - q),

(38)

Vc(I, J) - V2C(I, J) = ~6m(1 - k)q2(1 - q)(3q + 1) +~6kq(1 - q);

(39)

hence, if we set 1 A = ~rn(1 - k)q2(1 - q)2,

(40)

498

A.R. BOUFFETTE AND J. BOUFFETTE

we get the following significant relations: Vm/2vs(LJ)- VFs(LJ)=2[Vc(LJ)

- V ~ : z p s ( L J ) ] + A,

A

V z c ( L J ) - V v s ( I , J) = 2[ V c ( L J ) - V z c ( L J)] + ~-,

(41) (42)

which allows one to build Figure 1 representing the viability for some of the typical values of the coefficient of parentage. The previous diagram is only linear in the specific cases w h e n the coefficient A is null, that is, w h e n the frequency of the a gene is equal to 1 or 0 (in the case of h o m o g e n e o u s populations), or rn ---0 (in the case of no embryonic mortality), or also w h e n k = 1 (in the instance w h e n all the (aa) larva die before they can reach adult age). The coefficient A has a maximum value, equal to 1/64, w h e n the frequency of the a gene is equal to 1/2, only the embryonic mortality of the (aa) exists and acts systematically. In these conditions, the difference of viabilities of a b r o t h e r sister couple and a half-brother-sister couple is 3/64-4.7%, whereas the one of a cousins couple and a half-brother-sister couple is 1 / 6 4 - 1.6%. We can also prove the following inequalities: ~< Max( Vl/2ps - VFS) <~

5

(43)

~< Max( V c - V,/zvs) <~

(44)

~6~
(45)

V v s ) <~ 9

Vc

1 I I I I

m

0

Figure 1.

Viability and coefficient of parentage.

M A T H E M A T I C A L MODEL FOR THE STUDY OF FERTILITY

Viability and

Total Genetic Load.

499

The expression for the viability

V(L J), 9

V ( L J ) = 1 - k~px~q - k(1 - ~u)q 2 - m ~. At" at(q),

(46)

r=l

is to be related to the probability for a specific zygote to survive the deleterious effects of a given locus, 1 - q~ps - q2(1 - (p)s - 2q(1 - q)(1 - q~)sh,

(47)

where q is the frequency of the mutant deleterious allele, s is the probability of death for an homozygotous mutant, h is the measure of prevalence and ~ is the coefficient of inbreeding. This last expression is at the basis of the method of accumulation of the complete genetic load S proposed by Morton et al. (1956). Taking for granted that the effects of the various alleles and the environment are independent and additive, they consequently estimate the parameters A and B in the following expression for S. S = e-(A+BF).

(48)

The experimental values necessary to estimate the load are obtained by determining the number of survivors in inbreedings whose coefficient of inbreeding is known, such as brother-sister inbreedings, or cousins, etc. Actually, these experimental values are also values of viability and can thus be used to check our model. An analysis of the literature made it possible for us to obtain some values of viability for different types of inbreedings; the differences between viabilities corresponding to these inbreedings do not exceed the already established maxima. Moreover, it is interesting to remark that the model given by Morton et al. (1956) suggests an exponential evolution of S according to the coefficient of inbreeding and that we must expect to obtain experimental curves represented in Figures 2a and b. Yet, if the data we could obtain experimentally deviate more from the theoretical curves than from the ones giving the evolution of viability according to the coefficient of inbreeding (which is the case of the data found by Mettler and Gregg, 1969), we could be led to reformulate the notion of genetic load. The existence of an inbreeding sensitivity gene, whose expressivity itself is dependent upon cytoplasmic regulations, can, as a matter of fact, account for the totality of the inbreeding effects throughout development. The model of Morton et al. (1956) implies a multitude of genes with small

500

A.R. BOUFFETTE AND J. BOUFFETTE S

S J

e- (zl+8)

/

e-A

e-(,4+8)

CA

I

--q:,

I

B
(a)

B> 0

(b)

Figures 2 a, b. Genetic load and coefficient of parentage.

independent effects w h i c h a c c u m u l a t e several 'effects' being n e e d e d to cause death. Our model only implies one genetic u n i t y w h o s e expression is modulated by c y t o p l a s m i c factors. M o r e o v e r , it is e x p e r i m e n t a l l y possible to estimate k, m and the f r e q u e n c y of the b r o t h e r - s i s t e r couples aa × aa, from w h i c h we can derive the f r e q u e n c y q of the a gene in the population. K n o w i n g k, m and q for a given population, we will be able to easily derive an estimate of the viability for some other s y s t e m s of inbreeding made within similar experimental conditions.

LITERATURE Biemont, C. 1978. "Inbreeding effects: evidence for a genetic system which regulates viability in Drosophila melanogaster populations." Mech. Ageing Dev. 8, 21-42. , 1980. "An inbreeding sensitivity gene in Drosophila melanogaster." Experientia 36, 169-170. Dobzhansky, Th., B. Spassky and T. Tidwell. 1963. "Genetics of natural populations of Drosophila pseudoobscura." Genetics 48, 361-373. Gillois, M. 1965. "Relation d'identit6 en g~n6tique." Ann. Inst. Henri Poincar~ 2, 1-94. Jacquard, A. 1970. Structures G~n~tiques des Populations. Masson et Cie: Paris. Malecot, G. 1948. Les Math~matiques de l'H~r~dit& Masson et Cie: Paris. Malogolowkin-Cohen, C. H., H. Levene, N. P. Dobzhansky and A. S. Simmons. 1964. "Inbreeding and the mutational and balanced load in natural populations of Drosophila willistoni," Genetics 50, 1299-1311. Mettler, L. E. and T. G. Gregg. 1969. Population Genetics and Evolution. Prentice Hall: Englewood Cliffs. Morton, N. E., J. F. Crow and H. J. Muller. 1956. "An estimate of the mutational damage in man from data on consanguineous marriages." Proc. Natn Acad. Sci. U.S.A. 42,855-863. Stone, W. S., F. D. Wilson and V. L. Gerstenberg. 1963. "Genetic studies of natural populations of Drosophila pseudoobscura, a large dominant population." Genetics 48, 1089-1106. Toroja, E. 1964. "Genetics loads in irradiated experimental populations of Drosophila pseudoobscura." Genetics 50, 1289-1298. RECEIVED 10-22-80 REVISED 7-13-81