Multinomial-Sampling Models for Random Genetic Drift

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Multinomial-Sampling Models

for Random Genetic Drift

Thomas

Nagylaki

Department of Ecology and Evolution, The University of Chicago, Chicago, Illinois 60637 Manuscript received August 8, 1996

Accepted for publication October 15, 1996

ABSTRACT

Three different derivations of models with multinomial sampling of genotypes in a finite population are presented. The three derivations correspond to the operation of random drift through population regulation, conditioning on the total number of progeny, and culling, respectively. Generations are discrete and nonoverlapping; the diploid population mates at random. Each derivation applies to a

single multiallelic locus in a monoecious or dioecious population; in the latter case, the locus may be autosomal or X-linked. Mutation and viability selection are arbitrary; there are no fertility differences. In a monoecious population, the model yields the Wright-Fisher model ( i e . , multinomial sampling of genes) if and only if the viabilities are multiplicative. In a dioecious population, the analogous reduction does not occur even for pure random drift. Thus, multinomial sampling of genotypes generally does not lead to multinomial sampling of genes. Although the Wright-Fisher model probably lacks a sound biological basis and may be inaccurate for small populations, it is usually (perhaps always) a good approximation for genotypic multinomial sampling in large populations.

M

OST investigations of random drift in a finite population use the Wright-Fisher model or its extensions, either exactly or approximately. That model is based on the assumption that

genes

are sampled multinomially. Be-

cause of the ubiquity and importance of the Wright-Fisher model, it is disturbing that no convincing biological rationale appears to have been advanced for this assumption in diploids, and, excluding the neutral, monoecious case, it is far from obvious how to provide one.

ETHIER and NAGYLAKI (1980) formulated three types of models for random drift at a single multiallelic locus. One type involves overlapping generations and will not concern us here. The other two involve discrete, nonoverlapping generations, with either multinomial sampling or a general progeny distribution, and are directly relevant to this study. For each type of model, ETHIER

and NAG- examined both monoecious and dio-

ecious populations; in the latter case, they treated both autosomal and X-linked loci.

In their multinomial-sampling models, random drift acts through population regulation. This biologically explicit formulation leads to multinomial sampling of genotypes, which yields multinomial sampling of genes only in a monoecious population with multiplicative viabilities (ETHIER and NACYLAKI 1980; NAGYL,AKI 1986,

ETHIER and NAGYWU (1980) constructed their gen-

eral-progeny-distribution models by assuming first that

all matings produce offspring independently, with the same arbitrary progeny distribution, and then fixing 1990, 1992, pp. 248-253).

Address for correspondace: Thomas Nagylaki, Department of Ecology

and Evolution, The University of Chicago, 1101 E. 57th St., Chicago,

IL 60637.

the total number of offspring (separately for males and females in a dioecious population) by conditioning. They did not incorporate selection.

This paper has two related aims. First, by recapitulat- ing the pertinent aspects of the analysis of the mono-

ecious case (ETHIER and NACYLAKI 1980; NAGYWU

1986, 1990, 1992,

pp.

248-253) and extending those of the two dioecious cases, we shall explicate and em- phasize that population regulation generally does not lead to the Wright-Fisher model. Second, we shall present two additional derivations of genotypic multinomial sampling, both based on an underlying Poisson offspring distribution. In the first derivation, we use an extension of the general-progeny-distribution models of ETHIER and NAGYL.AKI (1980) that incorporates selection. In the second, we use new models in which random drift operates through culling.

It is important to note that the three derivations of our multinomial-sampling models are based on different sets of biological assumptions. Population regulation acts on an infinite population, whereas in the general-progeny-distribution and culling models, the population is fi-

nite at every stage of the life cycle. In the general-progeny- distribution models, conditioning restricts transitions to those with the desired, fixed total number of offspring. In the culling models, every transition in which the total number of offspring exceeds the desired number may occur; then culling reduces the population size to the fixed number. Only because of the profound role of the underlying Poisson offspring distribution do the three sets of assumptions lead to the same mathematical model.

Stochastic models for the genotypic frequencies en- able us to investigate exactly the evolution of both the allelic and genotypic frequencies, whereas gene-frequency models generally permit only the former.

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In the next three sections, we shall study successively a monoecious population, an autosomal locus in a d i e ecious population, and an X-linked locus. In each of these sections, we shall deduce our multinomial-sampling model from population regulation, conditioning on the total number of progeny, and culling. We shall summarize and discuss our results in the last section.

Generations are discrete and nonoverlapping; the d i p loid population mates at random. We focus attention on a single multiallelic locus. Mutation and viability selection are arbitrary, but there are no fertility differences.

MONOECIOUS POPULATION

First, we shall deduce our model from population regulation and discuss its relation to the Wright-Fisher model. Then we shall derive our model from the alternative hypotheses of fixed total offspring number and culling.

Population regulation: We summarize the relevant part of the development in NAGYIAKI (1992, pp. 248-253).

The life cycle starts with N monoecious adults. We focus attention on a single locus with multiple alleles Ai and denote the frequency of the unordered genotype Asl,, where i 5

J,

just before reproduction by Ptf It is

essential to use unordered frequencies as the basic vari- ables because random variation would destroy the sym- metry of the ordered frequencies. At this stage, the frequency of A , is

p i = P,,

+ x

c

Pi]

+

k

c

pl,. (1) I :I' I I : / < ,

Reproduction is panmictic, including selfing, and without fertility differences. The adults produce an infinite number of gametes, which fuse at random to form zygotes in Hardy-Weinberg proportions with unordered genotypic frequencies ( 2 -

6,)pipl,

in which

6,

represents the Kronecker delta (1 if i = j a n d 0 if i f

j)

and

p,

is still given by (1).

Selection acts through viability differences. If 7uij represents the viability of AtAj individuals, after selection the genotypic frequencies read

= ( 2 - 6tj)wzj*i*j/c ( 2 - 6kl)wk@kpl. (2) ksI

The population size remains infinite.

ity that Ai mutates to A,; by convention, ut, = 0. Then To introduce mutation, let utj designate the probabil-

&I = - u z k

6,

+

ug

(

( 3 )

is the probability that an Ai allele in a zygote appears as AI in a gamete. Assume that the two genes carried by an individual mutate independently. Consequently, after mutation the germ-line genotypic frequencies are

P;* = %(2 -

6,)

C

(&&/

+

&j&)Pg. (4)

k s 1

The population number is, of course, unaltered.

Random drift operates through population regulation, which reduces the population to N adults with unordered genotypic frequencies P>, thereby complet- ing the life cycle. Therefore, given the genotypic frequencies P, the distribution of the genotypic numbers

NP;, ( i 5 I ] is multinomial with index Nand parameters P f * . We shall write such statements concisely as

NP'

-

Y(N,

P**). (5)

We summarize the above information in the following formal scheme.

To some extent, the order of the evolutionary forces in our life cycle is arbitrary. We assume that while selection acts on the phenotype, which develops from the zygotic genotype, the germ cells mutate with no pheno- typic effect. Then in any formal scheme, selection must precede mutation. We are left with two possible se- quences in addition to ours: reproduction, selection, regulation, mutation; and reproduction, regulation, selection, mutation. Whereas in our model selection and mutation occur in an infinite population and hence can be treated deterministically, the first of the above alternatives would entail a much more complicated probabilistic formulation of mutation, and the second would necessitate this for both mutation and selection. The latter alternative is studied in the next subsection. It is convenient to express the transition probabilities (5) of the Markov chain of genotypic frequencies in terms of probability-generating functions:

Most of our interest centers on the evolution of the gene frequencies. Since the transition probabilities (5)

depend on the initial genotypic frequencies P only through the allelic frequencies p, therefore the vector of gene frequencies p ( t) , where t (=0, 1, 2,

-

) denotes time in generations, is Markovian. Hence, we can investigate the Markov chain (p(t)) without analyzing (P(t)

1;

the consequent decrease in dimensionality greatly simplifies many calculations. We now deduce the generating function of the transition probabilities of (p(t)).

Computations within generations are facilitated by em- ploying the ordered (or symmetrized) genotypic frequencies

Then (1) reduces to

(3)

with the definition wi, = wii for i

<

j , (2) becomes

P

f = w,,p;p,/a,

a

=

x

wqpip,; (9)

i, j

and (4) reads

P

f

*

=

&iRLF$.

(10)

k, 1

Summing (10) over j gives

p**

1 =

c

Rkzpt,

(11)

k

where

p

f = piwi/a, wi =

x

wqpj. (12) i

To derive the generating function of the transition probabilities of the Markov chain of gene frequencies, take

tii

= in (6) for every i and j such that i 5 j .

We find (NAGYLAKI 1986, 1992, p. 251)

Since the right-hand side depends only on p, we obtain

In the Wright-Fisher model, given p, the allelic numbers 2Npl are multinomially distributed with index 2N and parameters

p

T:*

2%'

-

.r)r(2N, p**). (15)

The corresponding probability-generating function is

(x

I

p " c j z N

=

(x

TP7*ei,)N.

(16)

1.1 . .

Since 1; is arbitrary, we see at once that (14) and (16) are equivalent if and only if

Pf*

=

pTp?

for every i

and j , that is, if and only if the adults are in Hardy- Weinberg proportions. This occurs if and only if the fitnesses are multiplicative, i.e., w,, = vivj for every i and

j for some v (NAGYLAKI 1992, pp. 76, 95).

The above proof of the sufficiency of multiplicative viabilities for the validity of the Wright-Fisher model is essentially the same as that of ETHIER and NAGYLAKI (1980). Their proof of necessity, however, is different and uses the conditional covariance between the gene frequencies.

Fixed total progeny number: Here, we incorporate selection into the general-progeny-distribution model of ETHIER and NAGYLAKI (1980) for a monoecious population.

Assume first that all the matings produce offspring independently, with the same progeny distribution, which has positive probabilities of producing 0 and 1 offspring. Then viability selection (introduced through survival probabilities) and mutation follow. Finally, condition to choose only those transitions in which the total number of progeny is N.

This model is formulated mathematically in the AP- PENDIX. KARLIN and MCGRECOR (1964,1968) proposed a similar model for asexual haploids. See also the diploid models of WATTERSON (1970).

In the APPENDIX, we prove that if the underlying offspring distribution is Poisson, then the transition probabilities of our model are given by the multinomial distribution (5).

Culling We make the same assumptions as in the last model, except that, instead of conditioning on a total offspring number of exactly N, we sample N adults without replacement from the juvenile offspring population,

i.e., after selection and mutation. Thus, if M signifies the number of juveniles, we condition only on M 2 N.

This is probably the most realistic of the three types of models studied in this paper. See GILLESPIE (1975) and NAGYLAKI (1995) for discussions of haploid and diploid culling models, respectively.

We now demonstrate that if the underlying offspring distribution is Poisson, then the transition probabilities of this model are given by the multinomial distribution (5). First, choose an integer m 2 Nand condition on

total juvenile offspring number M = m. According to (A8), if the underlying offspring distribution is Poisson, then the distribution of the juvenile genotypic numbers is multinomial with index m and parameters P$*:

Clearly, sampling

N

adults without replacement from these m juveniles yields a multinomial distribution with index N a n d the same parameters P f * , i.e., for each fixed m, we obtain the transition probabilities (5). Since this conditional distribution is independent of m, it is identical to its average over M , which is the desired transition probability.

AN AUTOSOMAL LOCUS IN A DIOECIOUS POPULATION

Here, we extend our investigation of a monoecious population to a dioecious one. First, we shall deduce our model from population regulation and establish that it never reduces to the dioecious Wright-Fisher model. Then we shall derive our model from the hy- pothesis of fixed total numbers of male and female offspring and from that of culling.

Population regulation: The superscript s signifies sex: s = 1 for males and s =

2

for females. The life cycle starts with

N,

adults of sex s, in whom the frequency of the unordered genotype As?, ( i 5 I ] is

e;).

At this stage,

the frequency of A, in sex s is

(4)

If wg) represents the viability of A&, individuals of sex

s, after selection the genotypic frequencies are given by

P;)* = w $ ) P p / z l w(Jp‘O) k.

.

(20)

Let u!;) designate the probability that A, mutates to A, in sex s; by convention, uif’ = 0. Mutation changes the genotypic frequencies to

P:;j** = %(2 -

6,)

(RpRjj’

+

R‘”’Rp)P:;’*, ki (21)

k s 1

where

RI;’ = 1 -

x

u y

6,

+

u ! A )

( h )

’ ’

(22)

After population regulation, the conditional distribution of the genotypic numbers in each sex is multinomial,

N,P(”’

-

-3((Ny, P(Sj**), (23)

and the genotypic numbers in males (NIP(’)’) and females (&P(‘)’) are mutually independent. In terms of probability-generating functions, we have

[.,!(

fJ

n

( t : ; ) ) Y q p ( l j , p‘2)) =

fJ

(c

*fi;)**$*

, = l ZSJ s = l L l j

(24)

The above model is due to ETHIER and NAGW

(1980). For two alleles, a similar model had been formulated earlier by MOWN (1958,1962, pp. 144-152), who introduced selection by assigning different constant fertilities to the three genotypes. If matings occur, however, even in an infinite population this formulation requires multiplicative fertilities (PENROSE 1949; BODMER

1965; NAGYLAIU 1992, pp. 47-51, 170); the necessary detailed biological assumptions in a finite population are unknown.

To investigate the evolution of the gene frequencies, note first that P ( ) * * depends only on the gene frequencies p“) and p(‘), which therefore form the Markov chain {p‘” ( t ) ,

P ‘ ~ ’

( t)

1.

To derive the generating function of the transition probabilities of this Markov chain, we first introduce symmetrized genotypic frequencies:

p y

=

X (

1

+

S,.)PI;”, 2 5

j ;

P(;j

I = P 5 ) . ‘I (25)

Defining wj:’ = wi;’ for i

<

j , from (18)

to (21) we obtain

pj” =

per,.

?I 7 (26)

p

= &(pjup;2) + $ ( I ) ( 2 )

3

pt

); (27)

p;j* =

w;;jp$o)/a(9,

ds)

=

w(:)p(!x

=

‘ I v

C

4;p i Pj >

3

5 ) (1) (2).

l.1 z, j

(28)

= Ri:)Rjj’Piij*. (29)

k, 1

Setting )<; =

51”5f’

in (24) leads to the generating

function

2

[,,(

n

( < y v , P : r ) ’

I

p ( l ) , p‘2’

)

=

l

j

(f;

Pf;)il’)S:r)**

)?

c = l 2

(30)

In the dioecious Wright-Fisher model, the male and female allelic numbers are independently and multinomially distributed:

2N5p“”

-

11(2N,, p‘r’**), (31)

which corresponds to the generating function

f,>(

n

2

n

( ~ ~ ~ ) ) 2 w q p ( ~ ) , p( 2i ) =

n

4

(E

p:l)**<:r)) 1 N <

.

r = l i , = I i

(32)

By the argument below (15), we see that (30) and (32) are equivalent if and only if

Pb<)**

= p?**pJ’j** (33)

for every i,

j ,

and s. But (33) is false even for pure random drift, when it is

pY;o,

= p ~ s ’ p j l ’ . (34)

p y

7

X(pp

+

pi“)

(35)

Recalling (27) and summing (34) over

j

gives

for every i and s, which is equivalent to

p:”

=

$I“

for every i. This cannot always hold in a stochastic model.

This result was first stated by ETHIER and NAGVLAIU (1980).

Fived total progeny number: Our assumptions are the same as in the corresponding model for a monoecious population, except for the following. In the reproduction step of the life cycle, we now posit also that in each mating, conditioned on the total number of progeny of that mating, the distribution of the number of male (or female) progeny is binomial. At the final stage of the life cycle, we condition to choose only those transitions in which the total number of progeny of sex s is

Nr.

This model is formulated mathematically in the AP- PENDIX. It differs from the corresponding model of

ETHIER and NAGYLAKI (1980) in two ways: in their

model, there is no selection, and mutation occurs during gametogenesis, whereas in the present model, selection is included and mutation occurs in the offspring, after selection. The two models agree if there is no selection and the mutation rates in males and females are equal. The special case of the ETHIER-NAGYLAKI model with two alleles and equal male and female mutation rates had been formulated and investigated earlier by FELDMAN (1966).

In the APPENDIX, we prove that if the underlying off-

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Culling: We make the same assumptions as in the last model, except that, instead of conditioning on exactly N, offspring of sex s, we sample N, adults without replacement from the juvenile offspring of sex s for s

= 1, 2. Thus, if M, signifies the number ofjuveniles of sex s, we condition only on Mr 2 N,.

We extend the corresponding proof in the monoecious case to demonstrate that if the underlying offspring distribution is Poisson, then the transition probabilities are given by the independent male and female multinomial distributions (23). First, choose integers ml and q such that m, 2 N, and condition on exactly

M, = m, offspring of sex s. According to ( A l l ) , if the underlying offspring distribution is Poisson, then the male and female offspring genotypic numbers have the independent multinomial distributions

Sampling N, adults of sex s without replacement from the m,juveniles yields (23) for each (ml, q). Since (23) is independent of ( m l , q), it is identical to its average over (Ml, M2), and is therefore the desired transition probability.

AN X-LINED LOCUS

Here, we study an X-linked locus by modifying the analysis of the dioecious autosomal case. All the results are analogous.

Population regulation: For females, (18) to (23) hold with s = 2. If wj') represents the viability of A,

males, then the male genotypic frequencies after selection are given by

= w y ) p y ) / a ( l ) ,

a(')

= wj1)p2(2).

(37)

1

Mutation changes these frequencies to

p p * *

= R p p y * . ( 3 8 )

3

After population regulation, we have

N1p""

-

.?1(N,, p(I)**); (39)

the genotypic numbers

Nip'"'

and N2P(')' are mutually independent. Instead of (24), the probability-generating function is now

This model is due to ETHIER and NAGYLAKI (1980). The gene frequencies [p'" ( t ) , p") ( t ) ] are again Mar- kovian. For s = 2, we symmetrize as in (25) to (29). Instead of (30), we obtain

In the Wright-Fisher model, one posits (39) for males and (31) for females, with independent allelic numbers

Nip'"'

and 2N2p(')'. This corresponds to the generating function

The argument below (32) holds for s = 2 and implies that (41) and (42) differ even for pure random drift, as stated by ETHIER and NAGYLAKI (1980).

Fixed total progeny number: The verbal descrip- tion above of the corresponding model for an autosomal locus in a dioecious population applies unaltered. The required mathematical modifications are in the

Culling: The treatment of the dioecious autosomal case requires only minor modifications. Instead of (36), from (A14) we now obtain the independent multinomial distributions

APPENDIX.

mlp(')'

-

Y(ml, p(')**), m2p(')'

-

~ ( q , P(~)**), (43)

and these lead to (39) for males and (23) for females.

DISCUSSION

Here, we summarize our main results and add some

We have derived genotypic multinomial-sampling models for random drift from three alternative hypotheses: population regulation, conditioning on the total progeny number, and culling. In the monoecious and dioecious autosomal cases, the models are given by (5)

and (23), respectively. For an X-linked locus, (39) applies to males and (23) to females.

The Wright-Fisher model and its extensions are based on the assumption that genes, rather than genotypes, are sampled multinomially. Our models are consistent with that assumption only in the monoecious case, and then if and only if viabilities are multiplicative. In no other situation has a convincing rationale been advanced for the Wright-Fisher model. This lacuna is important because disconfirmation of a biological assumption is more informative than that of an intuitive Ansatz.

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cal basis and may be inaccurate for small populations, it is usually (perhaps always) a good approximation for genotypic multinomial sampling in large populations. This justifies most evolutionary applications.

Part of this work was done during my visit to the Erwin Schrddinger Institute and the Institute for Mathematics, University of Vienna. I am most grateful for their hospitality.

LITERATURE CITED

BODMER, W. F., 1965 Differential fertility in population genetics. Genetics 51: 41 1-424.

ETHIER, S. N., and T. NACYLAKI, 1980 Diffusion approximations of

Markov chains with two time scales and applications to popuia- tion genetics. Adv. Appl. Probab. 12: 14-49.

FELDMAN, M. W., 1966 On the offspring number distribution in a

genetic population. J. Appl. Probab. 3 129-141.

GILLESPIE, J. H., 1975 Natural selection for within-generation vari- ance in offspring number. 11. Discrete haploid models. Genetics

KARLIN, S., and J. MCGRECOR, 1964 Direct product branching processes and related Markov chains. Proc. Natl. Acad. Sci. USA 51:

KARLIN, S., and J. MCGRECOR, 1968 The role of the Poisson progeny distribution in population genetic models. Math. Biosci. 2:

11-17.

MOW, P. A. P., 1958 A general theory of the distribution of gene frequencies. 11. Non-overlapping generations. Proc. Roy. SOC. B

MORAN, P. A. P., 1962 The Statistical Processes OfEvolutionaly Theory. Clarendon Press, Oxford.

NACYLAKI, T., 1986 The Gaussian approximation for random genetic drift, pp, 629-642 in Evolutionary Processes and Thsmy, edited by S. KARLIN and E. NEVO. Academic Press, New York.

NAGYLAKI, T., 1990 Models and approximations for random genetic

drift. Theor. Popul. Biol. 37: 192-212.

NAGYLAKI, T., 1992 Introduction to Theoretical Population Genetics. Springer-Verlag, Berlin.

NAGYLAKI, T., 1995 The inbreeding effective population number in a dioecious population. Genetics 139: 473-485.

PENROSE, L. S., 1949 The meaning of "fitness" in human populations. Ann. Eugen. 1 4 301-304.

WAITERSON, G. A,, 1970 O n the equivalence of random mating and

random union of gametes models in finite, monoecious populations. Theor. Popul. Biol. i: 233-250.

81: 403-413.

598-602.

149 113-116.

Communicating editor: W. J. EWENS

APPENDIX

Monoecious population: After formulating the model with fixed total progeny number, we shall establish that it reduces to genotypic multinomial sampling if the underlying offspring distribution is Poisson.

Assume first that all the matings produce offspring independently, with the same probability-generating function (p.g.f.) f ( 9 ) . An offspring with genotype A&] ( i

I 1) survives to adulthood with probability 7uz1, Mutation follows. Each offspring in a given mating obtains its genotype (one of four, each with probability survives, and mutates independently. Finally, condition to choose only those transitions in which the total number of progeny is M ( 2 1). Although here we are concerned

only with constant population number ( M = N ) , the generalization to arbitrary fixed M will be required for the analysis of the monoecious culling model. To en- sure a positive transition probability for each M , we

require that each mating produce 0 and 1 offspring with positive probabilities:

f ( 0 )

>

0 and

5

(0)

>

0. ('41)

We condition on

P

throughout. After mutation, a sur- viving ApAv offspring has genotype AJ, with probability

%(2

-

bL,) ( W L I

+

R&J.

Since there are N2PhPmn matings of type A& X A d n ,

the p.g.f. of the offspring genotypic numbers just before conditioning o n the total number of progeny is

x(()

=

n n

f

2l

c c

1 - %I,

k 5 1 r n s n p = k . l u = m , n

(

(A24

in which the argument offis the p.g.f. of the genotypic number of a single offspring from an AkAl X A,A, mating. From (A2a) we obtain the p.g.f. of the offspring

genotypic numbers whose sum is M.

$(!5 M ) = c"[%",

X ( % ) l >

(A2b) where the right-hand side denotes the coefficient of

BM

in

x

(e(),

and

(06)

= d l s . Therefore, I/J( 1, M ) gives the probability that the total offspring number is M ,

where each component of the vector 1 is 1. We con- clude that the p.g.f. of M" conditioned on both the paternal genotypic frequencies P and total offspring number M is

Suppose now that

f ( 6 ) = e + ' ) (A3)

the Poisson p.g.f., where K

>

0 is a constant (the mean

of the corresponding Poisson distribution). We reduce

( A 2 )

by transforming to the symmetrized genotypic frequencies P and then transforming back to P. First, we need two simple identities. If the quantities a,, are sym- metric, i.e., if a,,, = a,,,,! for every m and n, then

c

P m n f l m n =

c

~ ? , t 7 z % n n . 644)

7 " s 71 m. n

For any quantities

b,,,

we have

Prnn

b,, = 2

x

Pmnbpm.

(fi)

m,n u = m , n m, n

Successively using (A3), (A4), (A5), ( 8 ) , (9), ( l o ) ,

(7)

Substituting (A6) into (A2b) and the result into ( A ~ c ) , we obtain

M

('47)

whence

Mp'

-

. Y ( M , P**),

which agrees with (5) in the special case M = N.

An autosomal locus in a dioecious population: As

in the monoecious case, we first formulate the model with fixed total numbers of male and female offspring and then establish that it reduces to genotypic multinomial sampling if the underlying offspring distribution is Poisson.

Our assumptions are the same as in the last subsection, except for the following. In the reproduction step of the life cycle, we now posit also that, in each mating, conditioned on the total number of progeny of that mating, each offspring independently has sex s with probability T,,, where 0

<

7rl

<

1 and 7 r l

+

7 r 2 = 1. An offspring of sex s with genotype AJ1 ( i 5 y) survives to

adulthood with probability w?'. In the final step of the life cycle, we condition to choose only those transitions in which the total number of progeny of sex s is M ,

(21). The generalization from N = ( N 1 , N2) to M =

(MI, M 2 ) will be required for the analysis of the corresponding culling model.

A straightforward extension of the derivation of (A2) demonstrates that the conditional p.g.f. of the geno- rypic numbers is

where

In general, if there is selection, then

4

depends on 7 r s

and thus on the zygotic sex ratio 7r1/7r?. However, in

the absence of selection, 7 r 5 and 0, occur in

x

only in

the product 7r,@,, so

+

involves 7rTT, only in the product

7rYvrF2,

which cancels out of

4.

We now posit (A3) and follow the proof in the mono-

ecious case. Successively invoking (A4) and (A5) for each sex and then using (26) to (29) and (25) leads to

2

x

( p , 5 ' 2 ' ) = -KN1N? 7rJd') (1 - p!;)**<:;))]

,

1= 1 1 5 7

(A101

whence

2

+ & I ) , f 2 ) , M) =

n

F'!;)**[:;) , ( A l l )

Y

$ = I (is,

)

which agrees with (24) in the special case M = N. Note that, even with selection, in the Poisson case

4

is independent of T , ~ .

An X - l i e d locus: Here, we describe the modifications of the analysis and remarks in the last subsection for an autosomal locus in a dioecious population.

A male offspring with genotype Ai survives to adulthood with probability w : ' ) . Now (A9) becomes

4(&('),

!?), M)

(A1 2a)

where

(A1 2c)

Positing (A3) now leads to

whence

(-414)

Multinomial-Sampling Models for Random Genetic Drift