Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol166.Issue2.Aug2000:

The e€ective number of a population that varies cyclically in

size. I. Discrete generations

Yufeng Wang

a,1

, Edward Pollak

b,* a

Department of Zoology and Genetics, Iowa State University, Ames, IA 50011-1210, USA

b

Statistical Lab, Department of Statistics, 111 Snedecor Hall, Iowa State University, Ames, IA 50011-1210, USA

Received 28 September 1999; received in revised form 9 May 2000; accepted 16 May 2000

Abstract

We consider a dioecious population having numbers of males and females that vary over time in cycles of lengthk. It is shown that if kis small in comparison with the numbers of males and females in any gen-eration of the cycle, the e€ective population number (or size),Ne, is approximately equal to the harmonic mean of the e€ective population sizes during any given cycle. This result holds whether the locus under consideration is autosomal or sex-linked and whether inbreeding e€ective population numbers or variance e€ective population numbers are involved in the calculation of Ne. If, however, only two successive gen-erations in the cycle are considered and the population changes in size between these gengen-erations, the in-breeding e€ective population number,NeI, di€ers from the variance e€ective population number,NeV. The mutation e€ective population number turns out to be the same as the number derived using calculations involving probabilities of identity by descent. It is also shown that, at least in one special case, the ei-genvalue e€ective population number is the same as NeV. Ó _{2000 Published by Elsevier Science Inc. All}

rights reserved.

Keywords:E€ective population size; Cyclic variation

1. Introduction

Random genetic drift is an important in¯uence on the genetic variability of a ®nite population and a numerical measure of its in¯uence is the e€ective population size. Wright [1±3] presented the ®rst example for a dioecious population and a general expression for monoecious populations that

*_{Corresponding author. Tel.: +1-515 294 7765; fax: +1-515 294 4040.}

E-mail addresses:ywang@iastate.edu (Y. Wang), pllk@iastate.edu (E. Pollak).

1

Tel.: +1-515 294 9053.

do not change in size. Later, Crow [4] showed, for a monoecious population, that the e€ective population size measuring the increase in the probability that two copies of a gene are identical by descent di€ers from the e€ective population size involved in the variance of allele frequency changes, except if the size of the population remains constant. Other early work on this subject is discussed in the text by Crow and Kimura [5].

Recently, Caballero [6] and Nagylaki [7] derived general expressions for the e€ective population sizes of random mating dioecious populations, whether the locus under consideration is autos-omal or sex-linked. For an autosautos-omal locus, the results are consistent with those obtained by Crow and Denniston [8]. Caballero [6] and Nagylaki [7] assume, however, that the numbers of males and females do not change between generations. However, wild populations can ¯uctuate considerably in size from year to year. A simple way to describe such populations, which may not be too unrealistic, is to assume that these changes occur in repeated cycles. Wright [2,3] showed that, for monoecious populations in which individuals have approximate Poisson distributions of numbers of successful gametes in o€spring, the e€ective population size is approximately equal to the harmonic mean of the populations sizes in a cycle, provided the length of a cycle is short in comparison to those sizes. Pollak [9] has shown that this result also holds generally for o€spring distributions with ®nite variances. However, to the best our knowledge, no such results have yet been presented for dioecious populations. In the next two sections we shall derive e€ective pop-ulation sizes for autosomal and sex-linked loci, by generalizing the reasoning used by Caballero [6] and Nagylaki [7], whereby recurrence equations are obtained for probabilities of identity by descent. Next, there will be alternative derivations for autosomal and sex-linked loci, which rely on obtaining the variance of the change in the frequency of an allele in a generation. Here the reasoning will be a discrete generation version of the reasoning used by Hill [10] and Pollak [11], respectively, for an autosomal and a sex-linked locus. This alternative approach leads to the same expressions as before for e€ective population sizes if the population is followed through an entire cycle, although the individual terms in the resulting harmonic mean are not quite the same as the corresponding terms that arise from the identity-by-descent approach. Mutation e€ective

popula-tion numbers are also derived, as are eigenvalue e€ective populapopula-tion numbers in one special case. 2

2. Autosomal loci

We consider an autosomal locus in a population whose size undergoes repeated cycles of length

k. Let us suppose that at timestandt‡1 the population is respectively in generationsiandi‡1

of a cycle. Then, at timet‡1,

Fi‡1;t‡1 ˆthe inbreeding coefficient of a random individual

and

fvw;i‡1;t‡1 ˆthe coefficient of coancestry of a random pair of separate individuals of sexes

v and w;

2

wherevandwcan be replaced by either of the symbolsmandf, denoting, respectively, male and

because half of the copies of a gene in an individual come from a male and half come from a female. These equations can be simpli®ed if we set

Ui;tˆ1ÿFi;t;

dvw;itˆ1ÿfvw;i;t:

Then (1) and (2) reduce, respectively, to

Ui‡1;t‡1 ˆdmf;i;t …3†

Eqs. (3) and (4) can be rewritten in matrix notation as

We shall show later that if, at each stage i in the cycle, the numbers N…i† m and N

…i†

f of males and

females are large, all the elements P…i†

u;vw in the matrix Di are small. It can then be proved by

induction that

Ai‡rÿ1Ai‡rÿ2 Ai ˆ …A0‡Di‡rÿ1†…A0‡Di‡rÿ2† …A0‡Di† Ar0‡ X

rÿ1

jˆ0

Ar₀ÿ1ÿjDi‡jA j

0;rP1;

if rmax‰N…i† m ;N

…i†

f Š. Thus, if kmax‰Nm…i†;N …i† f Š,

di‡k;t‡k Ak0

"

‡X

kÿ1

jˆ0

Ak₀ÿ1ÿjDi‡jA j 0 #

di;tˆ ‰M‡DiŠdi;t; …6†

whereMˆAk₀ and the elements of

Di ˆ Xkÿ1

jˆ0

Ak₀ÿ1ÿjDi‡jA j 0

are all small. Now

Ar_o ˆ

0 1

4 1 2

1 4

0 1

4 1 2

1 4

0 1

4 1 2

1 4

0 1

4 1 2

1 4 2

6 6 6 4

3

7 7 7 5

ˆ

1

1 1

1 2

6 6 6 4

3

7 7 7 5

0 1

4 1 2

1 4

; rP2;

and it can also be shown that the eigenvalues ofAr₀arek1 ˆ1;k2 ˆk3 ˆk4 ˆ0. The left and right

eigenvectors p0 and v that correspond to k1ˆ1; and satisfy the normalization conditions

p0_vˆp0_{1ˆ1 are}

p0ˆ 0 1 4

1 2

1 4

;

and

vˆ1ˆ ‰1 1 1 1Š0:

Since all the elements ofDi are small, the dominant eigenvalue ofM‡Di is

qˆ1‡d;

where d is small. By a standard result from perturbation theory, as discussed, for example, by

dp0Div

Thus, after many cycles, the probabilities of non-identity of pairs of copies of a gene shrink at a

steady rate that is approximately equal to 1‡dper cycle, whered, the right-hand side of (7) is not

dependent oni. This rate may then be set equal to‰1ÿ1=…2Ne†Šk, so that

uv to be the number of successful gametes contributed by a parent of sexuin

generationj of a cycle to an o€spring of sexv. Since we are considering neutral alleles and there

areN…j†

Since there is a random mating, any pair of gametes in o€spring of sexesv and w coming from

parents of sex u is just as probable as any other pair, even if both gametes come from a single

and therefore, (9) and (10) are substituted in (8), we obtain

1

3. Sex-linked loci

The notation remains the same as in the previous section, but nowP…i†

m;mm ˆP …i†

m;mf ˆ0 because

males get their copies of a gene only from their mothers. Females get half their copies of a gene from parents of each sex. Therefore,

Fi‡1;t‡1 ˆfmf;i;t; …12†

As before, the recurrence equations can be simpli®ed if we set

Ui;t ˆ1ÿFi;t;

dvw;i;t ˆ1ÿfvw;it:

Eqs. (12)±(15) are then replaced by

where

It follows from (16) that

di‡k;t‡k Ak0

because, as in the previous section, it will be shown that all of the elements ofDi are small. The

characteristic equation corresponding toA0 is

jA0ÿkIj ˆ …ÿk†

r ofAr and the dominant eigenvalue 1 are, then,

p0ˆ 0 1

It follows from perturbation theory that if all the elements ofDi are small, the dominant

This is the same for alli. Thus

As (9) and (10) still hold for a sex-linked locus, all the elements ofDiare small. Their substitution in (18) leads to

4. Earlier results that are related to (11) and (19)

Let us assume thatP…j†

u;vwˆPu;vw,Nu…j† ˆNuandG…uvj†ˆGuvfor allj. There is then no cycle andkis replaced by 1 in (11) and (19). Expressions (11) and (19) then, respectively, reduce to

1

Another special case of (11) is where there are independent Poisson distributions of male and female o€spring of an individual in each generation. Then

Var…G…_uvj†† ˆN

…j‡1† v

Nu…j† and

Cov…G…_umj†;G…_ufj†† ˆ0:

Hence (11) reduces to

1

Ne

1

16k

Xkÿ1

jˆ0 1

Nm…j‡1† "

‡ 1

N_f…j‡1†‡

4

Nm…j†

ÿ 2

Nm…j‡1†

‡ 1

N_f…j‡1†‡

1

Nm…j‡1†

‡ 4

N_f…j†ÿ

2

N_f…j‡1†

#

ˆ 1

4k

Xkÿ1

jˆ0 1

Nm…j† "

‡ 1

N_f…j†

# ;

so that Ne is the harmonic mean of the ke€ective sizes in a cycle, each of which is of the form

derived by Wright [1].

Analogously, a special case of (19) is one for which there are independent Poisson distributions of male and female o€spring of a female and the number of daughters produced by a male has a Poisson distribution. Then

1

Ne

1

9k

Xkÿ1

jˆ0 1

N_f…j‡1†

"

‡ 1

Nm…j‡1†

‡ 4

N_f…j†ÿ

3

N_f…j‡1†‡

2

N_f…j‡1†‡

2

Nm…j†

ÿ 1

Nm…j‡1† #

ˆ 1

9k

Xkÿ1

jˆ0 4

N_f…j†

"

‡ 2

Nm…j† #

;

so thatNe is the harmonic mean of theke€ective population sizes in a cycle, each of which is of

the form derived by Wright [15].

5. Another characterization ofN_e

In this section, we will express the results given by (11) and (19) in terms of variance e€ective numbers within a cycle. The reasoning will be based on that used by Hill [10]. Thus, we take account of three sources of variability when gametes are transmitted from one generation to the next: between numbers of gametes produced by di€erent individuals, between genotypes of in-dividuals, and among alleles carried by gametes when parents are heterozygous.

We consider a population in which parents and o€spring are, respectively, in phasesjandj‡1

of a cycle. LetG…j†

uvr be the number of successful gametes contributed by therth parent of sexuto

o€spring of sexv, and

X_ur…j†ˆthe frequency of allelleB1 in the rth parent of sexu;

d…_uvr`j† ˆthe difference betweenX_ur…j† and the frequency of B1 in the

`th gamete contributed to an offspring of sexv by the rth parent of sex u:

homozygote or a male with an X-linked locus and 1=2 with equal probabilities if it is a het-erozygote.

If there is an autosomal locus the male and female o€spring, respectively, originate from 2N…j‡1† m

and 2N_f…j‡1† successful gametes. These two sets of gametes are drawn from a population whose

frequency ofB1 isp. The frequencyp0ofB1 among the o€spring is the unweighted average of the

frequencies among males and females. Thus

p0ˆ 1

As we are considering neutral alleles the distribution ofG…j†

uvr is the same for allr, so thatG

After some algebra, it follows from Eqs. (22)±(25) that

Var…p0jp† ˆE‰…p0ÿp†2jpŠ

Since the changes in allele frequencies in di€erent generations are independent,N…j†

m †, the variance of the total change in allele frequency

throughout a cycle is approximately equal to Pk

jˆ1p…1ÿp†=…2N …j†

eV†. Hence, we can de®ne the

e€ective population size to be

1

Note that the individual expressions being summed in (11) di€er, with respect to terms not in-volving variances and covariances, from corresponding terms in (27). However, the sum of such expressions in (11) is

Xkÿ1

so that (11) and (27) give identical expressions for the e€ective population number.

Now let us suppose that there is a sex-linked locus. In this case the N_f…j‡1† female o€spring

originate from 2N_f…j‡1† successful gametes, but the N_m…j‡1† males are derived from only N_m…j‡1†

ga-metes, all of which were contributed by their mothers. Another new feature is that the frequencyp0

among the o€spring is a weighted average, with weights 1/3 and 2/3 for males and females. Thus

Eqs. (24) and (25) still hold if uˆf, but now

Var…X…j†

mr† ˆp…1ÿp†; …29†

because the frequencies of B1Y and B2Y are pand 1ÿp.

Eq. (23) also remains valid except for the fact that E…G…j†

mmr† ˆ0. It therefore follows from

Eqs. (23)±(25) and (29) that

Var…p0jp† ˆE‰…p0ÿp†2jpŠ

The terms in (19) that do not involve variances and covariances add to

Xkÿ1

so that, while individual summands in (19) and (31) di€er, Ne, given by (19), and NeV, given by

(31), are the same.

6. Inbreeding and variance e€ective population numbers

If we assume that all the probabilitiesP…j†

u;vw are small and setduv;0;0 ˆ1 for allu andv, Eqs. (4) imply that

dmm;i‡1;t‡1 dff;i‡1;t‡1 dmf;i‡1;t‡114‰dmm;it‡2dmf;i;t‡dff;i;tŠ

for all t. Thus, if we rede®ne the time scale so that tˆ0 andtˆ2 respectively indicate times at

U2;2 ˆdmf;11 1

8 P

…0† m;mf h

‡P_f;mf…0† iU0;0‡ 1

ÿ1

4 P

…0† m;mf h

‡P_f…0†_;mfi

dmf;00

ˆ1

8 P

…0† m;mf h

‡P_f;mf…0† iU0;0‡ 1

ÿ1

4 P

…0† m;mf h

‡P_f…0†_;mfi

U1;1: …32†

This equation is a generalization of the usual recurrence equation connecting three successive values of the panmictic index when there are independent Poisson distributions of numbers of

male and female o€spring of an individual. It thus makes sense to replace 1

4N …0† m ‡14N

…0†

f in that

special case by 1

NeI

ˆ1

4 P

…0† m;mf h

‡P_f…0†_;mfi; …33†

which is the probability that both gametes uniting to produce an individual in generation 2 came from the same grandparent in generation 0. If we simplify our notation, by setting

E G …0†_uvˆN

…1† v

Nu…1†

ˆl_uv

and

CovG…0†_um;G…0†_ufˆrum;uf;

it follows, when (10) is substituted in (33), that

1

NeI

ˆ1

4

rmm;mf ‡lmmlmf

Nm…0†l_mml_mf

"

‡rfm;ff ‡lfmlff

N_f…0†l_fml_ff #

; …34†

as found by Crow and Denniston [8].

In the particular use in which rum;uf ˆ0, (34) reduces to

1

NeI

ˆ 1

4Nm…0†

‡ 1

4N_f…0†;

which would hold, for example, if each individual had independent Poisson distributions of

numbers of male and female o€spring. But then (27) implies, fork ˆ1, that

1

NeV

ˆ1

4 1

Nm…1† "

‡ 1

N_f…1†

# :

In general, (34) seems to be di€erent from the special case of (11) that results whenkis set equal

to 1. But the two approaches that are used to calculate NeI can be reconciled as follows. For

simplicity, we set

VarÿG…0†_uvˆr2_uv

and

r2_uˆVar G…0†_um

‡G…0†_uf

Next, we note that, given G…0†

It, therefore, follows from Eqs. (34) and (35) that

because

1

Nu…0†

l_uf l_um…l_um‡l_uf†

"

‡ lum

l_uf…l_um‡l_uf†‡

2 l_um‡l_uf

#

ˆ 1

Nu…0†

l_uf ‡l_um l_uml_uf " #

ˆ 1

Nu…0† 1 l_um "

‡ 1

l_uf #

ˆ 1

Nm…1†

‡ 1

N_f…1†:

Expression (36) is consistent with (11).

If a locus is sex-linked and we set kˆ1 and jˆ0

1

NeI

ÿ 1

NeV

1

9 4

1

N_f…0†

"

ÿ 1

N_f…1†

!

‡2 1

Nm…0†

ÿ 1

Nm…1† #

:

Suppose, in particular, that there are independent Poisson distributions of male and female o€-spring of a female, and the number of daughters produced by a male has a Poisson distribution. Hence, by the discussion in Section 4,

1

NeI

1

9 4

N_f…0†

"

‡ 2

Nm…0† #

and

1

NeV

1

9 4

N_f…1†

"

‡ 2

Nm…1† #

:

7. Mutation e€ective population numbers

Let P…i†

u;vw have the same meaning as in Sections 2 and 3. But we now assume that the in®nite

alleles model holds, so that genes mutate at a rate u, in such a way that each mutant is to an

entirely novel allelic type. Suppose that a population is in generationi‡1 of a cycle at timet‡1.

Then, at this time,Gi‡1;t‡1 andgvw;i‡1;t‡1 respectively denote the probabilities that two copies of a

gene in one individual, and in two random separate individuals of sexesvand w, are identical in

state. Then, if the locus under consideration is autosomal, Eqs. (1) and (2) are replaced by

Gi‡1;t‡1 ˆ …1ÿu†2gmf;it; …37†

and

gvw;i‡1;t‡1 ˆ

…1ÿu†2

8 P

…i† m;vw h n

‡P_f;vw…i†

i

‰1‡Gi;tŠ ‡2‰1ÿPm;vw…i† Šgmm;i;t

‡4gmf;i;t‡2‰1ÿP… i† f;vwŠgff;i;t

o

: …38†

If we now setHi‡1;t‡1 ˆ1ÿGi‡1;t‡1 and hvw;i‡1;t‡1 ˆ1ÿgvw;i‡1;t‡1, (37) and (38) imply that

and

hvw;i‡1;t‡1ˆ1ÿ …1ÿu†2‡

…1ÿu†2

8 P

…i† m;vw h n

‡P_f;vw…i† iHi;t‡2 1 h

ÿP_m;vw…i† ihmm;i;t‡4hmf;i;t

‡2 1h ÿP_f;vw…i† ihff;i;t o

: …40†

Then ifhi;t ˆ ‰Hi;t;hmm;i;t;hmf;i;t;hff;i;tŠ0, Eqs. (39) and (40) can be written in matrix notation as

hi‡1;t‡1 ˆ ‰1ÿ …1ÿu† 2

Š1‡ …1ÿu†2Aihi;t; …41†

whereAiis the same matrix as in Eq. (5) and1ˆ ‰1 1 1 1Š0. We assume thatuis very small. Hence, it follows from (6) and (41) and the fact that the sum of the elements of each row of any power of A0 is 1, that

hi‡k;t‡kˆ ‰I‡ …1ÿu† 2

Ai‡kÿ1‡ ‡ …1ÿu†2…kÿ1†Ai‡kÿ1 Ai‡1Š‰1ÿ …1ÿu† 2

Š1

‡ …1ÿu†2kAi‡kÿ1 Aihi;t2ku1‡ …1ÿ2ku†…M‡Di†hi;t:

Thus, if ckmax‰N…i† m ;N

…i† f ;uÿ1Š,

hi‡ck;t‡ck2cku1‡ …1ÿ2cku†qc1p0hi;t 2cku1‡ 1

ÿck 4Neu‡1

2Ne

1p0hi;t; …42†

where 1 and p0 _{are the eigenvectors associated with the dominant eigenvalue 1 of}

Ar₀;qˆ ‰1ÿ1=…2Ne†Šk, andNeis given by (8) and (11). The stationary value ofhi;t ishi, where, by (42),

hi ck 2u

ÿ 4Neu‡1

2Ne

p0hi

‡p0hi

1:

Hence, all the elements of hi are equal to Hi and, because p01ˆ1,

HiˆH

4Neu 4Neu‡1

; iˆ1;2;. . .;k: …43†

The deviations from equilibrium areei;t ˆhi;tÿhi and, by (42),

ei‡ck;t‡ck 1

ÿck 4Neu‡1

2Ne

1p0ei;t;

which shows that the equilibrium given by (43) is stable.

If the locus under consideration is sex-linked, it can be shown that

hi‡1;t‡1 ˆ ‰1ÿ …1ÿu† 2

Š1‡ …1ÿu†2Aihi;t; …44†

where Ai is now the same matrix as in (18). It then follows from the same sequence of steps as

those leading from (41) to (43) that

hi!Hi1

4Neu 4Neu‡1

1; …45†

whereNe is given by (19). Therefore, whether the locus is autosomal or sex-linked,

Gi;t!Gi

1 4Neu‡1

Thus,Ne, the mutation e€ective population number, as de®ned by Ewens [16], is the same as the inbreeding e€ective population number.

8. Eigenvalue e€ective population numbers

In what follows, let G…i†

uvr have the same meaning as in Section 5. Suppose that this random

variable has a binomial distribution withN…j‡1†

v trials and a probability 1=N

…j†

u of success. In

ad-dition, we assume thatG…j†

umr andG …j†

ufr are independently distributed if there is an autosomal locus

or when there is a sex-linked locus anduˆf. Then

Var…G…j† uvr† ˆ

N…j‡1†

v

Nu…j† 1

ÿ 1

Nu…j†

:

If a locus is autosomal (27) reduces to

1

NeV

1

4k

Xkÿ1

jˆ0 1

Nm…j‡1† "

‡ 1

N_f…j‡1†

#

; …46†

ifN…j‡1† m and N

…j‡1†

f are large for all j‡1. If a locus is sex-linked G…mmrj† 0 and we assume that

Var…G…_mfrj†† ˆN

…j‡1† f

Nm…j† 1

ÿ 1

Nm…j†

;

Eq. (31) implies that

1

NeV

1

9k

Xkÿ1

jˆ0 4

N_f…j‡1†

"

‡ 2

N_f…j‡1†

#

; …47†

if the numbers of males and females are always large.

Under the assumptions we have made about population sizes the covariance between the outputs of successful gametes from two separate individuals of the same sex is approximately 0. It is then also consistent with the special cases in the foregoing paragraph to consider the

distri-bution of copies of B1 passed on to adults of the next generation by parents of sex u to be

bi-nomial. LetYu…t‡1†be the number of copiesB1 among fertilized eggs that lead to adults of sexv in generationt‡1, and Xuv…t†be the number of copies of this allele passed on to these fertilized

eggs by parents of sexu. Then, if there is an autosomal locus

Yv…t‡1† ˆXmv…t† ‡Xfv…t†; …48†

whereXmv…t†andXfv…t†have independent binomial distributions. IfNu andNu0 respectively denote

the number of adults of sexu in generationt and t‡1

Xuv…t† Bin Nv0;

Yu…t† 2Nu

: …49†

It follows from (48) and (49) that

E Yv…t‡1†

2N0 v

Yu…t†;u

ˆm;f

ˆYm…t†

2Nm

‡Yf…t†

2Nf

E Yv…t‡1†

If, therefore, we average the right-hand sides of Eqs. (50)±(52) over the joint distributions ofYm…t† and Yf…t† and setmv…t† ˆE‰Yv…t†=…2Nv…t††Š and muv…t† ˆE‰Yu…t†Yv…t†=…4NuNv†Š; we obtain the sys-tem of equations

mt‡1 ˆT…t†mt; …53†

The characteristic equation ofT…t†can be shown to be

jT…t† ÿkIj ˆk2…kÿ1† k2

This equation has a single root equal to 1 and two others equal to 0. Among the remaining roots the one of largest absolute value is

k…t‡1† ˆ1

If the numbers of males and females do not change between generations this expression reduces to

If we now follow the population over a whole cycle, (53) leads to

mt‡kˆT…t‡kÿ1†T…t‡kÿ2† T…t†mt:

Thus, if there ares cycles

msk A (

‡B Y k

jˆ1 k…j†

" #s)

m0; s! 1; …54†

whereAand Bare matrices that depend on the phase of the cycle at time 0. Note that, by (46),

Yk

jˆ1

k…j† 1ÿ1

8 Xk

jˆ1 1

Nm…j† "

‡ 1

N_f…j†

#

1ÿ k

2NeV

so that at the level of approximations used in this paper, the recurrence equations on the ®rst and second moments lead to the variance e€ective population number.

We claim that, at least for the model discussed in this section, NeV is the same as NeE, the

ei-genvalue e€ective size of Ewens [17,18], which is the largest non-unit eiei-genvalue of the matrix of transition probabilities of numbers of copies of B, in males and females when the same phase is

considered in two successive cycles. If second moments are traced throughscycles, andsis large,

they each satisfy

muv…sk†

1 2

Ym…0†

Nm…0† "

‡Yf…0†

N_f…0†

#

‡ 1

ÿ 1

2NeE ks

muv…sk jAC†; …55†

wheremuv…skjAC† is the expected value of the uvth moment, given the asymptotic conditional

distribution of allele frequency pairs in males and females whenB1has been neither ®xed nor lost.

Since (54) and (55) are of the same form,NeE is at least approximately equal to NeV.

If the locus under consideration is sex-linked (48) still holds ifvˆf, but

Xmf…t† Bin N_f0;

Ym…t†

Nm

and

Ym…t‡1† ˆXfm…t† Bin Nm0;

Yf…t† 2Nf

:

Again, we assume that any pair of the random variables Xuv…t† are independent. It can then be

shown that

mt‡1ˆT…t†mt;

wheremm…t† ˆYm…t†=Nm;mm…t‡1† ˆYm…t‡1†=Nm0 and

T…t† ˆ

0 1 0 0 0

1 2

1

2 0 0 0

0 1

Nm0 0 0 1ÿ

1 Nm0

0 0 0 1

2

1 2 1

4N_f0 1 4N_f0

1 4…1ÿ

1 N_f0†

1 2

1 4…1ÿ

1 N_f0† 2

6 6 6 6 6 6 6 4

3

The characteristic equation corresponding toT…t† is

k2

ÿ1

2kÿ

1 2

(

ÿk3‡k2 3

4ÿ

1 4N0

f !

‡k 3

8ÿ

1 4N0

m

ÿ 1

8N0 f

‡ 1

4N0 mNf0

!

ÿ1

8 1ÿ

1

N0

m

ÿ 1

N0

f

‡ 1

N0

mNf0 !)

ˆ k2

ÿ1

2kÿ

1 2

f…k† ˆ0:

The ®rst factor has roots 1 and ÿ1

2, whereas

f…1† ÿ 1

8N0 m

‡ 1

4N_f0

! ;

f…0†<0;f 1

2

ˆ1

8; f

ÿ1

2

ˆ 1

8N0 f

‡ 1

4N0 m

;

and

f0…1† ˆ ÿ9

8‡O

1

N0

m ; 1

N0

f !

:

Hence, by using Newton's method and the intermediate value theorem, we ®nd that the largest root of f…0† ˆ0 is approximately

1ÿ 1

9N0 m

‡ 2

9N_f0

!

and the other two roots are between 0 and1

2in magnitude, with one being negative and the other

positive. Thus, if we trace the population throughscycles we ®nd, as for an autosomal locus, that

1

NeE

1

9k

Xk

yˆ1 2

Nm…j†

‡ 4

N_f…j†

!

1

NeV

:

We have not yet been able to generalize the foregoing reasoning to arbitrary o€spring

distri-butions, but conjecture that then alsoNeENeV. Another reason to suspect that this is the case is

that Eq. (31), with k ˆ1, was ®rst derived by Pollak [19] by a computation of the approximate

rate at which the probabilities in the asymptotic conditional distribution decrease. This method

also led to Eq. (27) when kˆ1. For either an autosomal or a sex-linked locus the branching

process approximation used by Pollak [19] involved looking forward in time from parents to o€spring, so that a variance e€ective population number was obtained.

9. Discussion

and is the same regardless of whether the harmonic mean is calculated from inbreeding e€ective numbers or variance e€ective numbers. We obtain the same result by calculating the mutation e€ective size, and, as far as we can tell, the eigenvalue e€ective size. Of course, as shown in Section

6, the inbreeding e€ective number,NeI, and the variance e€ective number,NeV, are not the same if

only two successive generations are considered and the population changes in the size. As far as we know, such expressions have not previously been given when there is a sex-linked locus.

In our calculations, we have not restricted ourselves to assuming Poisson distributions of o€-spring. Thus complete account has been taken of both ¯uctuations in population size and vari-ability in family sizes. Frankham [20] reviewed 192 published estimates from 102 species of the

ratio of Ne to the actual population size, N, and concluded that the ®rst and second most

im-portant variables explaining variation among these estimates were, respectively, ¯uctuation in

population size and variance in family size. These variables both act to reduceNe=N to the low

values found by him in wildlife populations.

References

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[2] S. Wright, Size of population and breeding structure in relation to evalution, Science 87 (1938) 430.

[3] S. Wright, Statistical genetics in relation to evolution, in: Actualities scienti®ques et industrielles, vol. 802, Exposes de Biometrie et de la statistique biologique, XIII, Hermann, Paris, 1939, p. 5.

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