Fixation Indices in Subdivided Populations

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Copyright1998 by the Genetics Society of America

Fixation Indices in Subdivided Populations

Thomas Nagylaki

Department of Ecology and Evolution, The University of Chicago, Chicago, Illinois 60637

Manuscript received April 30, 1997 Accepted for publication October 3, 1997

ABSTRACT

Without restricting the evolutionary forces that may be present, the theory of fixation indices, or F-sta-tistics, in an arbitrarily subdivided population is developed systematically in terms of allelic and genotypic frequencies. The fixation indices for each homozygous genotype are expressed in terms of the fixation indices for the heterozygous genotypes. Therefore, together with the allelic frequencies, the latter suffice to describe population structure. Possible random fluctuations in the allelic frequencies (which may be caused, e.g., by finiteness of the subpopulations) are incorporated so that the fixation indices are parameters, rather than random variables, and these parameters are expressed in terms of ratios of evolutionary expectations of heterozygosities. The interpretation of some measures of population differentiation is also discussed. In particular, FSTis an appropriate index of gene-frequency differentiation if and only if the genetic diversity is low.

W

RIGHT’s fixation indices, or F-statistics, are the pectations or probabilities.Nei’s indices, however,

be-parameters most widely used to describe popula- come random variables through their dependence on tion structure.Wright (1969, pp. 294–295; 1978, pp. the allelic and genotypic frequencies in the population.

80–89; and refs. therein) defined the fixation indices Therefore, his indices are more difficult to relate to as correlations between uniting gametes. His treatment theoretical investigations of population structure (

Nagy-is restricted to neutral diallelic loci; it Nagy-is somewhat artifi- laki1989 and refs. therein;Nagylakiet al. 1993), which

cial (because numerical values are assigned to gametes) are usually formulated in terms of covariances of allelic and not entirely clear. frequencies or probabilities of identity in allelic state or

Cockerham(1969, 1973; see alsoWeirandCocker- _{of identity by descent.}

ham1984;CockerhamandWeir1986) based his study _{Here, we shall combine some of the desirable}

proper-of population structure on the analysis proper-of the variance ties of the treatments ofCockerham andNei. In the

and covariances of indicator variables for allelic state, next section, we shall developNei’s approach fully and

and he related his parameters to fixation indices and systematically for deterministic genotypic frequencies. measures of identity by descent. AlthoughCockerham’s _{Then we shall extend our analysis to randomly varying}

analysis is more lucid and general thanWright’s, it is _{allelic frequencies. In the final section, we shall discuss}

disturbing that negative variance components may oc- _{some of our results and the interpretation of some} mea-cur if mates are less closely related than the average _{sures of population differentiation.}

within subpopulations (i.e., ifWright’s F_IS, 0). Nei (1973; 1977; 1986; 1987, pp. 159–166; see also

Nei and Chesser 1983) presented a third approach,

DETERMINISTIC G ENOTYPIC FREQUENCIES

formulated entirely in terms of the allelic and genotypic

After defining the fixation indices, we shall present frequencies in the population. He expressed the

fixa-the constraints fixa-they satisfy, express fixa-the indices for each tion indices in terms of ratios of heterozygosities. His

homozygote in terms of the indices for heterozygotes, treatment is biologically the most direct, and it clearly

derive the generalization ofWright’s hierarchical

rela-requires no restrictions on the action of the evolutionary

tionship among the indices, and evaluate the comple-forces.

ment of each index as a ratio of heterozygosities. Allelic and genotypic frequencies may fluctuate

be-The population is subdivided into an arbitrary num-cause of finite subpopulation numbers or random

varia-tion in evoluvaria-tionary forces. Even in this case,Wright’s ber of subpopulations. Let w

kdenote the proportion of

andCockerham’s measures of population structure are the population in subpopulation k, so that

still parameters because they are defined in terms of

ex-o

_k wk5 1. (1)

We consider a single autosomal locus with r alleles Ai.

Address for correspondence: Thomas Nagylaki, Department of Ecology

The frequencies of the allele Aiand the ordered

geno-and Evolution, The University of Chicago, 1101 East 57th Street,

Chicago, IL 60637. type AiAjin subpopulation k are pi,kand Pij,k, respectively.

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Thus, Pij,k 5 Pji,k for every i and j, and the frequencies

FST,ii5

Var(pi) pi(12pi)

, (10a)

of the unordered genotypes AiAiand AiAjin

subpopula-tion k are Pii,k and 2Pij,k for i≠j, respectively. Then we

have FST,ij5 2

Cov(pi,pj) pipj

i≠ j, (10b)

pi,k5

o

j

Pij,k. (2)

and noting that

The frequencies of the allele Ai and the genotype AiAj Var(pi)#pi(12 pi) (11a)

in the entire population are

and pi5

o

k

wkpi,k, Pij5

o

k

wkPij,k, (3)

[Cov(pi,pj)]2#Var(pi) Var(pj)#pi(12 pi)pj(1 2pj),

(11b) where the bar indicates averaging over subpopulations.

We do not restrict the action of the evolutionary _{from (10) we infer} forces, except that they must be deterministic. This

im-0# FST,ii# 1 (12a)

plies, in particular, that every subpopulation must be

(in principle) infinite. ₍Chakraborty1993) and

We now define Nei’s (1977) genotype-specific

fixa-tion indices. The subscripts I, S, and T refer to individu- _|_F_ST,ij_|_#

3

(12 pi)(1 2pj) pipj

4

1/2

, i≠j. (12b) als, subpopulations, and the total population,

respec-tively. The parameters FIS,ij,kand FIT,ijdesignate standardized _{Now we express the fixation indices for each}

homozy-measures of the deviation from Hardy-Weinberg pro- _{gote in terms of the heterozygote indices, which} there-portions of genotype AiAjin subpopulation k and in the _{fore suffice for the analysis of population structure.}

Sub-entire population, respectively; FST,ijsignifies a standard- _{stituting (4) into (2) leads to}

ized measure of the covariance of the frequencies of

(1 2pi,k)FIS,ii,k5

o

j :j≠ipj,kFIS,ij,k, (13)

the alleles Aiand Aj:

Pii,k5 pi,k2 1 FIS,ii,kpi,k(12pi,k), (4a) _{which can be rewritten more compactly but less}

instruc-tively as Pij,k5 (12FIS,ij,k) pi,kpj,k, i ≠j; (4b)

FIS,ii,k5

o

j

pj,kFIS,ij,k. Pii5pi2 1FIT,iipi(12pi), (5a)

Pij 5(12 FIT,ij)pipj, i≠j; (5b)

Inserting (5) into the average of (2) yields (National

p2

i 5pi2 1FST,iipi(1 2pi), (6a) Research Council 1996, Appendix 4A)

pipj 5(12 FST,ij)pipj, i≠j. (6b) (1 2pi)FIT,ii5

o

j :j≠i

pjFIT,ij. (14)

If every subpopulation is panmictic, then (4) implies

Finally, substituting (6) into the equation that FIS,ij,k50 for every i, j, and k. In this case, Pij5pipj,

so comparing (5) with (6) informs us that FIT,ij5 FST,ij

o

j

pipj5 pi,

for every i and j.

we find that FST,ijalso satisfies (14):

The panmictic indices are the complements of the fixation indices:

(12 pi)FST,ii5

o

j :j≠i

pjFST,ij. (15)

HIS,ij,k5 12 FIS,ij,k, (7a)

Thus, in each subpopulation, the1_⁄2_{r (r}₁₁₎₂₁

inde-HIT,ij5 12 FIT,ij, (7b)

pendent genotypic frequencies can be replaced by the HST,ij5 12 FST,ij. (7c) r21 independent allelic frequencies and the1⁄2r(r21)

heterozygote fixation indices FIS,ij,k(i≠j). An analogous

The fixation indices satisfy some simple constraints.

reparametrization holds for the mean genotypic fre-From (4b), (5b), and (6b) we see immediately

quencies in (5) and the covariances [see (10)] in (6). Note that if FIS,ij,k5 F˜IS,k, independent of i and j, for FIS,ij,k,FIT,ij, FST,ij# 1, i≠j. (8)

every i and j such that i ≠ j, then (13) appropriately These fixation indices can be negative. Since 0 # Pii,k _{implies that F}

IS,ii,k5F˜IS,k for every i. Similar results hold

#pi,kand 0#Pii#pi, from (4a) and (5a) we conclude _{for F}

IT,ijand FST,ij.

Next, we derive the generalization of Wright’s

2 pi,k

12pi,k

#FIS,ii,k# 1, 2 pi

12 pi

#FIT,ii#1, (9) (1943) relationship among the fixation indices. First,

guided by (4), we define the weighted average of FIS,ij,k

which is misprinted inChakraborty(1993). Rewriting over subpopulations (Nei 1977; Wright 1978, pp.

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hS,k512fS,k5

o

i,j :i≠jpi,kpj,k5

o

i

pi,k(12pi,k), (23b) FIS,ii5

1 pi2 pi2

o

k

wkpi,k(12 pi,k)FIS,ii,k, (16a)

hS512fS5

o

i,j :i≠jpipj5

o

i

(pi2pi2)5

o

k

wkhS,k. (23c) FIS,ij5

1 pipj

o

k

wkpi,kpj,kFIS,ij,k, i≠ j. (16b)

Therefore, fS,kis the probability that two genes chosen

Inserting (8) into (16b) and (9) into (16a) demonstrates

at random from subpopulation k are the same allele; that FIS,ij# 1 for every i and j. Since the averages (16)

the probability that two genes chosen at random from are properly normalized (i.e., the sum of the weights is

the same subpopulation are the same allele is fS. The

1), from (7a) we have

corresponding probabilities that the two genes are dif-ferent alleles are hS,kand hS.

HIS,ij512FIS,ij. (17)

If the entire population were panmictic, its homozy-Note carefully that the weighting in (16) differs from _{gosity and heterozygosity would become f}

T and hT,

re-that in (3). _spectively:

Solving (4) for FIS,ij,k, substituting into (16), and

recall-fT 5

o

i

p2

i , (24a)

ing (3), we deduce (Nei1977)

hT5 12 fT5

o

i,j :i≠j

pipj5

o

i

pi(1 2pi). (24b) FIS,ii5

Pii2pi2 pi2 pi2

, (18a)

Therefore, fT is the probability that two genes chosen FIS,ij5

pipj 2Pij pipj

, i≠j. (18b) _{at random from the entire population are the same}

allele; the probability that they are different alleles is We insert (13) into (16a) and invoke (16b) to express _h_T_{. From (23a) and (24a) we see at once that f}_S_$ _f_T_, every average homozygote index in terms of the average _{whence h}_S_# _h_T_.

heterozygote indices: _{We shall indicate averages over genotypes by an} aster-isk. Consider first FIS,ij,k. Multiplying (13) by pi,kand sum-FIS,ii5

1 pi2pi2j :j

o

≠i

pipjFIS,ij. (19) ming over i yields the equivalent homozygote and

het-erozygote averages Now we can prove that

F*IS,k 5

1 hS,k

o

i

pi,k(12pi,k)FIS,ii,k (25a) HIT,ij5 HIS,ijHST,ij (20)

for every genotype AiAj. From (18) we obtain ₅ 1 hS,ki,j :i

o

≠j

pi,kpj,kFIS,ij,k, (25b) Pii5pi21FIS,ii(pi2pi2), (21a)

which are properly normalized because of (23b). In-Pij 5(12 FIS,ij)pipj, i≠j. (21b)

serting (4b) into (25b) and invoking (22b) and (23b) For i 5 j, we equate (21a) to (5a), solve for FIT,ii, and leads to

invoke (6a), (7b), (7c), and (17) to establish (20). For

H*IS,k 512F *IS,k5hI,k/hS,k (26) i ≠ j, we equate (21b) to (5b), employ (7b) and (17),

solve for HIT,ij, and deduce (20) from (6b) and (7c). _{in every subpopulation k. Therefore, F*}

IS,kcan be negative,

Finally, we express each panmictic index as a ratio of _{but F *}

IS,k# 1 for every k .

heterozygosities, or gene diversities. Let fI,kand fIdenote _{Recalling (23c), we define the averages of F*} IS,k over

the actual homozygosities in subpopulation k and in _{subpopulations as} the entire population, respectively; the corresponding

heterozygosities are hI,kand hI: _F* IS5

1 hS

o

k

wkhS,kF *IS,k. (27) fI,k5

o

i

Pii,k, fI5

o

i

Pii5

o

k

wkfI,k, (22a)

Substituting (26) into (27) and employing (22c) and hI,k512fI,k5

o

i,j :i≠jPij,k, (22b) (23c) yields

H *IS5 12 F*IS 5hI/hS. (28) hI 512fI5

o

i,j :i≠jPij 5

o

k

wkhI,k. (22c)

This simple result, in which the numerator and denomi-If subpopulation k were panmictic, its homozygosity nator in (26) are averaged separately, follows from the would be fS,k; if every subpopulation were panmictic, the weightings in (25) and (27). Note that F *IScan be

nega-homozygosity in the entire population would be fS. The tive, but F *IS#1.

corresponding heterozygosities are hS,kand hS. Thus, By substituting (25) into (27) and appealing to (16),

we can also express F *ISas an average over homozygotes fS,k5

o

i p2

i,k, fS5

o

i

p2

i 5

o

k

wkfS,k, (23a)

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(28), (31), and (33) are ratios of random heterozygosi-F *IS5

1 hS

o

i

(pi 2pi2)FIS,ii (29a)

ties, even their expectations are difficult to evaluate and to relate to theoretical studies of population structure, which are usually formulated in terms of covariances of

5 1

hSi,j :i

o

≠j

pipjFIS,ij, (29b)

allelic frequencies or probabilities of identity in allelic state or of identity by descent. The fixation indices we which are properly normalized by (23c).

shall define are parameters. Now we turn to FIT,ij. Multiplying (14) by piand

sum-We shall examine only the allelic frequencies. These ming over i gives the equivalent homozygote and

hetero-are of greatest evolutionary interest and suffice for most zygote averages

theoretical investigations of population structure, which are usually restricted to panmictic subpopulations. To F *IT5

1 hT

o

i

pi(1 2pi)FIT,ii (30a)

account for random variation, we imagine that the popu-lation T, which comprises the subpopupopu-lations S, is

repli-5 1

hT i,j :i≠j

o

pipjFIT,ij, (30b) _{cated infinitely many times to form the metapopulation} U. Each of these replicates is an independent realization whose normalization is justified by (24). Inserting (5b) _{of the evolutionary process, so U is an infinite collection} into (30b) and utilizing (22c) and (24b), we obtain _{of such realizations. We do not assume that the}

subpopu-lations S are panmictic. H*IT5 12 F*IT5 hI/hT. (31)

The arrangement of this section is the same as that Therefore, F*IT# 1, but F *ITcan be negative. _{of the preceding one.}

For FST,ij, from (15) we get _{The allelic frequencies p}

i,kare now random variables.

As in the last section, a bar indicates averages over sub-F *ST 5

1 hT

o

i

pi(1 2pi)FST,ii (32a) _{populations S within the population T :}

pi 5

o

k

wkpi,k. (35)

5 1

hT ij:i≠j

o

pipjFST,ij. (32b)

Of course, piis now a random variable. For typographical

Substituting (6b) into (32b) and using (23c) and (24b),

simplicity, we use an angle bracket to signify averages we find

over evolutionary realizations (or sample paths). Thus, H *ST 512F*ST5 hS/hT. (33) kp_i,klis averaged over T within U, and the grand mean

of the frequency of Ai is

Since hT$ hS$ 0, we have 0#F *ST# 1.

From (28), (31), and (33) we infer at once the hierar- _p_i_; _E(p_i₎₅ _k_p_i_l. ₍₃₆₎ chical formula

Analogy with (21), (5), and (6) suggests the defini-H*IT 5H*ISH *ST. (34) tions

Nei(1977) derived (28), (31), (33), and (34) for homo- k

p2

il5 kpi2l1 FST,iikpi(12pi)l, (37a)

zygotes. Our treatment establishes these results also for

kpipjl5 (12FST,ij)kpipjl, i≠j; (37b)

heterozygotes. Observe from (34) that when (20) is

aver-aged over genotypes, the factors on the right-hand side kp2

il5 p2i 1FSU,iipi(12 pi), (38a)

are averaged separately. This occurs because the _k

pipjl5 (12FSU,ij)pipj, i≠ j; (38b)

weightings in (30) and (32) differ from those in (29).

kp2

il5 p2i 1FTU,iipi(12 pi), (39a)

In the above analysis, we posited a discretely

subdi-vided population. However, if we restrict our attention _k_p

ipjl5 (12FTU,ij)pipj, i≠j. (39b)

to FIT,ij, this assumption becomes unnecessary. Indeed,

As in (7), the panmictic indices are the complements the definitions (5), (22c), and (24) involve only allelic

of the above fixation indices. and genotypic frequencies in the entire population.

Solving (37) to (39) for the fixation indices yields Therefore, (14), (30), and (31) hold for arbitrary

popu-lation structure.

FST,ii5 kVar(p i|T )l

kpi(12 pi)l

, (40a)

STOCHASTIC ALLELIC FREQUENCIES

FST,ij5 2

kCov(pi, pj|T)l

kpipjl

, i≠j ; (40b) Here, we shall extend the analysis in the last section to

randomly varying allelic frequencies, which may reflect

FSU,ii5

Var(pi)

pi(12 pi)

, (41a)

finite subpopulation numbers or random variation in evolutionary forces. In this case, it is obvious thatNei’s

(1977) definitions (4), (5), and (6) lead to fixation _F

SU,ij5 2

Cov(pi,pj)

pipj

(5)

If the metapopulation U were panmictic, its homozygos-FTU,ii5

Var(pi)

pi(1 2 pi)

, (42a) ity and heterozygosity would be

fU5

o

i

p2

i, (49a)

FTU,ij5 2

Cov(pi, pj)

pipj

, i≠j . (42b)

hU5 12 fU5

o

i,j :i≠jpipj5

o

i

pi(12 pi). (49b)

A glance at (37b), (38b), and (39b) immediately

re-Note that the definitions (47), (48), and (49) follow veals the constraints

from the transformation of (22), (23), and (24), respec-FST,ij,FSU,ij,FTU,ij#1, i≠j. (43) _tively.

From (47a), (48a), and (49a) we obtain easily fS $

These fixation indices can be negative. Reasoning as in

fT$fU, which implies that hU# hT#hS.

(11), from (40a), (41a), and (42a) we deduce

To average FST,ijover homozygotes or heterozygotes,

0#FST,ii,FSU,ii,FTU,ii# 1. (44) _{we transform (29):}

Bounds corresponding to (12b) are easy to derive, but

F *ST5

1 hT

o

i

kpi(1 2pi)lFST,ii (50a)

are too complicated to be illuminating.

We can easily derive the remaining results in this

sec-tion ab initio, but we can obtain them more quickly by ₅ 1 hTi,j :i≠j

o

kpipjlFST,ij, (50b)

the following transformation. In (21), (5), and (6), we drop the bar from FIS,ij; make the substitutions I → S,

for which (28) yields S→ T, and T→ U; replace the bars by angle brackets;

and finally substitute Pij→pipjand pi→pi. This transfor- H *ST512F*ST 5hS/hT. (51)

mation yields pipj→ kpipjl and pi →pi. Then (21), (5),

For FSU,ij, from (30) and (31) we obtain

and (6) become (37), (38), and (39), respectively. To express the fixation indices for each homozygote

F *SU5

1 hU

o

i

pi(12 pi)FSU,ii (52a)

in terms of the heterozygote indices, we apply our trans-formation to (19), (14), and (15), which become,

re-spectively, ₅ 1

hUi,j :i

o

≠j

pipjFSU,ij. (52b)

FST,ii5

1

kpi(1 2pi)lj :j

o

≠i

kpipjlFST,ij, (45a) H *SU512F *SU5hS/hU. (53)

For FTU,ij, from (32) and (33) we get FSU,ii5

1 12 pij :j

o

≠i

pjFSU,ij, (45b)

F*TU5

1 hU

o

i

pi(12 pi)FTU,ii (54a) FTU,ii5

1 12 pij :j

o

≠i

pjFTU,ij. (45c)

5 1

hUi,j :i

o

≠j

pipjFTU,ij, (54b)

The generalization (20) of Wright’s relationship

H*TU512F *TU5hT/hU. (55)

among the fixation indices becomes

Since hS $ hT $ hU$ 0, the results (51), (53), and HSU,ij5HST,ijHTU,ij (46)

(55) inform us that for every i and j.

0# F*ST,F *SU,F*TU# 1,

Finally, we express each panmictic index as a ratio of expected heterozygosities. If every subpopulation S were

which also follows easily from (44), (50a), (52a), and panmictic, the expected homozygosity and

heterozygos-(54a). ity in the entire population T would be fSand hS,

respec-From (51), (53), and (55) we establish immediately tively. Thus, in this case, fSand hSare the homozygosity

the hierarchical result and heterozygosity in the metapopulation U:

H *SU5H *STH *TU, (56) fS5

o

i

kp2

il, (47a)

in accordance with (34).

The panmictic index H*ST is a measure of variation hS5 12 fS5

o

i,j :i≠jkpipjl5

o

i

kpi2 pi2l. (47b)

between subpopulations. Our development justifies the use of (51) for this parameter in theoretical investiga-If the entire population T were panmictic, these

expec-tions (see, e.g.,Takahata1983;CrowandAoki1984;

tations would become

TakahataandNei1984; Slatkin andBarton1989;

fT5

o

i

kp2

il, (48a) _Slatkin

1991, 1993), and the ratio (51) of expected heterozygosities may also be preferable for data analysis hT 512fT5

o

i,j :i≠j

kpipjl5

o

i

kpi(12 pi)l. (48b)

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heterozygosi-ties (NeiandChakravarti1977;Neiet al. 1977). Sub- population. Below, we develop this idea more precisely

and illustrate it by four examples. stituting (47) and (48) into (51) produces the explicit

Since nucleotide diversities are generally low, there-formula

fore F*STis usually a suitable measure of differentiation

at the nucleotide or codon level. H*ST5

12

o

i

kp2

il

12

o

i

kp2

il

. (57)

We separate the cases of high and low genetic diversity and use the criteria ofKimuraandMaruyama(1971);

√

uw /s. (67) Example 3: Our third example is the island model

For weak mutation (u ! 1) and large neighborhood (Moran1959;Maruyama1970;Maynard Smith1970;

size (Ns @ 1), the probability at equilibrium that two

Nagylaki 1983, 1986, and refs. therein). Generations

distinct genes sampled from demes separated by a dis-are discrete and nonoverlapping. Each of n ($2)

pan-tance w ($0) are the same allele is adequately approxi-mictic (including selfing) subpopulations comprises N

mated by (Nagylaki1989, and refs. therein)

monoecious, diploid individuals. These colonies ex-change gametes with no spatial effect on dispersion, i.e.,

f(j)≈ e2j

11 b, (68) if the migration rate is m (0 , m , 1), every colony

receives a proportion m/(n 2 1) of its gametes from

each of the other colonies. Selection is absent, and every whereb 54Ns

√

u designates a dimensionless parame-allele mutates to new parame-alleles at the same rate u (0 # ter. We set

u #1).

h(j)512f(j). (69)

We posit that migration is weak and that mutation is

The expected heterozygosity weak relative to the stronger one of migration and

ran-dom drift:

h(0)≈ b

11 b. (70) m! 1 and u!max(m ,1 /N). (63)

Then, at equilibrium, is high ifb *1 and low if b !1.

Now consider two demes with scaled separation j. ne≈

n[m1 u(4mNT1n2 1)]

nm1(n21)u (64) The effective number of alleles in these two demes is ne5

2 f (0)1 f(j)≈

2(11 b)

11e2j , (71) (Nagylaki1983), where N

T5nN represents the total

population number;

so their diversity is high ifb @1 and low ifb &1. For high diversity, we use f(j)/f (0) as a simple index F *ST≈

1

4Nma 11, (65) _{of differentiation between the two demes. Therefore,} differentiation is strong if e2j!1 and weak if e2j≈1, where a 5[n /(n21)]2₍_Nei_{1975, p. 123;}_Nagylaki

independent ofb. For low diversity, the measure h(0)/ 1983;Takahata1983;CrowandAoki1984;Takahata

h(j) reveals that differentiation is strong if andNei1984;CockerhamandWeir1987); and

differ-entiation is strong if and only if _{b !}₁₂_e2j _(72a)

4mN!max(1,4NTu) (66a) _{and weak if}

and weak if and only if _{b @} ₁₂ _e2j_. _(72b)

4mN@max(1,4NTu) (66b) From (61) we obtain

(Nagylaki 1986). Using F*_ST to assess differentiation

F*ST(j)5

h(j)2 h(0) h(j)1h(0) ≈

11e2j

11 2b 2 e2j. (73) would replace (66a) and (66b) by 4mN!1 and 4mN@

1, respectively, which is correct if and only if 4NTu#1.

Again, F*STyields the correct criterion for differentiation

Thus, F*STprovides the correct criterion for

differentia-if and only differentia-if diversity is low. tion if and only if diversity is low (cf. Nagylaki 1983,

I thankBrian Charlesworth, James F. Crow,_andMagnus Nord-1986).

borg_{for useful comments on the manuscript. This work was} sup-Example 4:Our last example is the unbounded,

unidi-ported by National Science Foundation grant DEB-9706912.

mensional stepping-stone model (Male´ cot1949, 1950,

1951;Kimura1953;Nagylaki1989, and refs. therein).

As in the island model, generations are discrete and

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