Fixation Indices in Subdivided Populations

Corresponding author: Thomas Nagylaki, Department of Ecology and Evolution, The University of Chicago, 1101 East 57th Street, Chicago, IL 60637.

Communicating editor: R. R. HUDSON

	ABSTRACT

TOP ABSTRACT DETERMINISTIC GENOTYPIC... STOCHASTIC ALLELIC FREQUENCIES DISCUSSION LITERATURE CITED

Without restricting the evolutionary forces that may be present, the theory of fixation indices, or F-statistics, 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, F_ST is an appropriate index of gene-frequency differentiation if and only if the genetic diversity is low.

WRIGHT's fixation indices, or F-statistics, are the parameters most widely used to describe population structure. WRIGHT 1969 (pp. 294–295; WRIGHT 1978 , pp. 80–89; and refs. therein) defined the fixation indices as correlations between uniting gametes. His treatment is restricted to neutral diallelic loci; it is somewhat artificial (because numerical values are assigned to gametes) and not entirely clear.

COCKERHAM 1969 , COCKERHAM 1973 (see also WEIR and COCKERHAM 1984 ; COCKERHAM and WEIR 1986 ) based his study of population structure on the analysis of the variance and covariances of indicator variables for allelic state, and he related his parameters to fixation indices and measures of identity by descent. Although COCKERHAM's analysis is more lucid and general than WRIGHT's, it is disturbing that negative variance components may occur if mates are less closely related than the average within subpopulations (i.e., if WRIGHT's F_IS < 0).

NEI 1973 ; NEI 1977 ; NEI 1986 ; NEI 1987 (pp. 159–166; see also NEI and CHESSER 1983 ) presented a third approach, formulated entirely in terms of the allelic and genotypic frequencies in the population. He expressed the fixation indices in terms of ratios of heterozygosities. His treatment is biologically the most direct, and it clearly requires no restrictions on the action of the evolutionary forces.

Allelic and genotypic frequencies may fluctuate because of finite subpopulation numbers or random variation in evolutionary forces. Even in this case, WRIGHT's and COCKERHAM's measures of population structure are still parameters because they are defined in terms of expectations or probabilities. NEI's indices, however, become random variables through their dependence on the allelic and genotypic frequencies in the population. Therefore, his indices are more difficult to relate to theoretical investigations of population structure (NAGYLAKI 1989 and refs. therein; NAGYLAKI et al. 1993 ), which are usually formulated in terms of covariances of allelic frequencies or probabilities of identity in allelic state or of identity by descent.

Here, we shall combine some of the desirable properties of the treatments of COCKERHAM and NEI. In the next section, we shall develop NEI's approach fully and systematically for deterministic genotypic frequencies. Then we shall extend our analysis to randomly varying allelic frequencies. In the final section, we shall discuss some of our results and the interpretation of some measures of population differentiation.

	DETERMINISTIC GENOTYPIC FREQUENCIES

TOP ABSTRACT DETERMINISTIC GENOTYPIC... STOCHASTIC ALLELIC FREQUENCIES DISCUSSION LITERATURE CITED

After defining the fixation indices, we shall present the constraints they satisfy, express the indices for each homozygote in terms of the indices for heterozygotes, derive the generalization of WRIGHT's hierarchical relationship among the indices, and evaluate the complement of each index as a ratio of heterozygosities.

The population is subdivided into an arbitrary number of subpopulations. Let w_k denote the proportion of the population in subpopulation k, so that

(1)

We consider a single autosomal locus with r alleles A_i. The frequencies of the allele A_i and the ordered genotype A_i A_j in subpopulation k are p_i,k and P_ij,k , respectively. Thus, P_ij,k = P_ji,k for every i and j, and the frequencies of the unordered genotypes A_i A_i and A_i A_j in subpopulation k are P_ii,k and 2P_ij,k for i j, respectively. Then we have

(2)

The frequencies of the allele A_i and the genotype A_i A_j in the entire population are

(3)

We do not restrict the action of the evolutionary forces, except that they must be deterministic. This implies, in particular, that every subpopulation must be (in principle) infinite.

We now define NEI's (1977) genotype-specific fixation indices. The subscripts I, S, and T refer to individuals, subpopulations, and the total population, respectively. The parameters F_IS,ij,k and F_IT,ij designate standardized measures of the deviation from Hardy-Weinberg proportions of genotype A_iA_j in subpopulation k and in the entire population, respectively; F_ST,ij signifies a standardized measure of the covariance of the frequencies of the alleles A_i and A_j :

(4a)

(4b)

(5a)

(5b)

(6a)

(6b)

If every subpopulation is panmictic, then (4) implies that F_IS,ij,k = 0 for every i, j, and k. In this case, _ij = , so comparing (5) with (6) informs us that F_IT,ij = F_ST,ij for every i and j.

The panmictic indices are the complements of the fixation indices:

(7a)

(7b)

(7c)

The fixation indices satisfy some simple constraints. From (4b), (5b), and (6b) we see immediately

(8)

These fixation indices can be negative. Since 0 P_ii,k p_i,k and 0 _{ii i}, from (4a) and (5a) we conclude

(9)

CHAKRABORTY 1993

(10a)

(10b)

(11a)

(11b)

(12a)

(CHAKRABORTY 1993 ) and

(12b)

Now we express the fixation indices for each homozygote in terms of the heterozygote indices, which therefore suffice for the analysis of population structure. Substituting (4) into (2) leads to

(13)

Inserting (5) into the average of (2) yields (NATIONAL RESEARCH COUNCIL 1996 , Appendix 4A)

(14)

Finally, substituting (6) into the equation

(15)

Thus, in each subpopulation, the ¹/₂r (r + 1) - 1 independent genotypic frequencies can be replaced by the r - 1 independent allelic frequencies and the ¹/₂r (r - 1) heterozygote fixation indices F_IS,ij,k (i j ). An analogous reparametrization holds for the mean genotypic frequencies in (5) and the covariances [see (10)] in (6).

Note that if F_IS,ij,k = F˜_IS,k , independent of i and j, for every i and j such that i j, then (13) appropriately implies that F_IS,ii,k = F˜_IS,k for every i. Similar results hold for F_IT,ij and F_ST,ij .

Next, we derive the generalization of WRIGHT 1943 relationship among the fixation indices. First, guided by (4), we define the weighted average of F_IS,ij,k over subpopulations (NEI 1977 ; WRIGHT 1978 , pp. 80–81):

(16a)

(16b)

Inserting (8) into (16b) and (9) into (16a) demonstrates that _IS,ij 1 for every i and j. Since the averages (16) are properly normalized (i.e., the sum of the weights is 1), from (7a) we have

(17)

Note carefully that the weighting in (16) differs from that in (3).

Solving (4) for F_IS,ij,k , substituting into (16), and recalling (3), we deduce (NEI 1977 )

(18a)

(18b)

We insert (13) into (16a) and invoke (16b) to express every average homozygote index in terms of the average heterozygote indices:

(19)

Now we can prove that

(20)

(21a)

(21b)

For i = j, we equate (21a) to (5a), solve for F_IT,ii , and invoke (6a), (7b), (7c), and (17) to establish (20). For i j, we equate (21b) to (5b), employ (7b) and (17), solve for H_IT,ij, and deduce (20) from (6b) and (7c).

Finally, we express each panmictic index as a ratio of heterozygosities, or gene diversities. Let f_I,k and _I denote the actual homozygosities in subpopulation k and in the entire population, respectively; the corresponding heterozygosities are h_I,k and _I :

(22a)

(22b)

(22c)

If subpopulation k were panmictic, its homozygosity would be f_S,k ; if every subpopulation were panmictic, the homozygosity in the entire population would be _S . The corresponding heterozygosities are h_S,k and _S . Thus,

(23a)

(23b)

(23c)

Therefore, f_S,k is the probability that two genes chosen at random from subpopulation k are the same allele; the probability that two genes chosen at random from the same subpopulation are the same allele is _S . The corresponding probabilities that the two genes are different alleles are h_S,k and _S.

If the entire population were panmictic, its homozygosity and heterozygosity would become f_T and h_T , respectively:

(24a)

(24b)

Therefore, f_T is the probability that two genes chosen at random from the entire population are the same allele; the probability that they are different alleles is h_T . From (23a) and (24a) we see at once that _S f_T , whence _S h_T .

We shall indicate averages over genotypes by an asterisk. Consider first F_IS,ij,k . Multiplying (13) by p_i,k and summing over i yields the equivalent homozygote and heterozygote averages

(25a)

(25b)

(26)

Recalling (23c), we define the averages of F^*_IS,k over subpopulations as

(27)

Substituting (26) into (27) and employing (22c) and (23c) yields

(28)

This simple result, in which the numerator and denominator in (26) are averaged separately, follows from the weightings in (25) and (27). Note that ^*_IS can be negative, but ^*_IS 1.

By substituting (25) into (27) and appealing to (16), we can also express ^*_IS as an average over homozygotes or heterozygotes:

(29a)

(29b)

Now we turn to F_IT,ij . Multiplying (14) by _i and summing over i gives the equivalent homozygote and heterozygote averages

(30a)

(30b)

(31)

Therefore, F^*_IT 1, but F^*_IT can be negative.

For F_ST,ij, from (15) we get

(32a)

(32b)

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

(33)

Since h_{T S} 0, we have 0 F^*_ST 1.

From (28), (31), and (33) we infer at once the hierarchical formula

(34)

NEI 1977 derived (28), (31), (33), and (34) for homozygotes. Our treatment establishes these results also for heterozygotes. Observe from (34) that when (20) is averaged over genotypes, the factors on the right-hand side are averaged separately. This occurs because the weightings in (30) and (32) differ from those in (29).

In the above analysis, we posited a discretely subdivided population. However, if we restrict our attention to F_IT,ij , this assumption becomes unnecessary. Indeed, the definitions (5), (22c), and (24) involve only allelic and genotypic frequencies in the entire population. Therefore, (14), (30), and (31) hold for arbitrary population structure.

	STOCHASTIC ALLELIC FREQUENCIES

TOP ABSTRACT DETERMINISTIC GENOTYPIC... STOCHASTIC ALLELIC FREQUENCIES DISCUSSION LITERATURE CITED

Here, we shall extend the analysis in the last section to randomly varying allelic frequencies, which may reflect finite subpopulation numbers or random variation in evolutionary forces. In this case, it is obvious that NEI's (1977) definitions (4), (5), and (6) lead to fixation indices that are random variables. Indeed, since (26), (28), (31), and (33) are ratios of random heterozygosities, 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 allelic frequencies or probabilities of identity in allelic state or of identity by descent. The fixation indices we shall define are parameters.

We shall examine only the allelic frequencies. These are of greatest evolutionary interest and suffice for most theoretical investigations of population structure, which are usually restricted to panmictic subpopulations. To account for random variation, we imagine that the population T, which comprises the subpopulations S, is replicated infinitely many times to form the metapopulation U. Each of these replicates is an independent realization of the evolutionary process, so U is an infinite collection of such realizations. We do not assume that the subpopulations S are panmictic.

The arrangement of this section is the same as that of the preceding one.

The allelic frequencies p_i,k are now random variables. As in the last section, a bar indicates averages over subpopulations S within the population T :

(35)

Of course, _i is now a random variable. For typographical simplicity, we use an angle bracket to signify averages over evolutionary realizations (or sample paths). Thus, p_i,k is averaged over T within U, and the grand mean of the frequency of A_i is

(36)

Analogy with (21), (5), and (6) suggests the definitions

(37a)

(37b)

(38a)

(38b)

(39a)

(39b)

As in (7), the panmictic indices are the complements of the above fixation indices.

Solving (37) to (39) for the fixation indices yields

(40a)

(40b)

(41a)

(41b)

(42a)

(42b)

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

(43)

These fixation indices can be negative. Reasoning as in (11), from (40a), (41a), and (42a) we deduce

(44)

Bounds corresponding to (12b) are easy to derive, but are too complicated to be illuminating.

We can easily derive the remaining results in this section ab initio, but we can obtain them more quickly by the following transformation. In (21), (5), and (6), we drop the bar from _IS,ij ; make the substitutions I S, S T, and T U; replace the bars by angle brackets; and finally substitute P_ij and p_{i i} . This transformation yields and _{i i}. Then (21), (5), and (6) become (37), (38), and (39), respectively.

To express the fixation indices for each homozygote in terms of the heterozygote indices, we apply our transformation to (19), (14), and (15), which become, respectively,

(45a)

(45b)

(45c)

The generalization (20) of WRIGHT's relationship among the fixation indices becomes

(46)

Finally, we express each panmictic index as a ratio of expected heterozygosities. If every subpopulation S were panmictic, the expected homozygosity and heterozygosity in the entire population T would be _S and _S , respectively. Thus, in this case, _S and _S are the homozygosity and heterozygosity in the metapopulation U:

(47a)

(47b)

If the entire population T were panmictic, these expectations would become

(48a)

(48b)

If the metapopulation U were panmictic, its homozygosity and heterozygosity would be

(49a)

(49b)

Note that the definitions (47), (48), and (49) follow from the transformation of (22), (23), and (24), respectively.

From (47a), (48a), and (49a) we obtain easily _S f_T f_U , which implies that h_U h_{T S} .

To average F_ST,ij over homozygotes or heterozygotes, we transform (29):

(50a)

(50b)

(51)

(52a)

(52b)

(53)

For F_TU,ij , from (32) and (33) we get

(54a)

(54b)

(55)

Since _S h_T h_U 0, the results (51), (53), and (55) inform us that

From (51), (53), and (55) we establish immediately the hierarchical result

(56)

The panmictic index H^*_ST is a measure of variation between subpopulations. Our development justifies the use of (51) for this parameter in theoretical investigations (see, e.g., TAKAHATA 1983 ; CROW and AOKI 1984 ; TAKAHATA and NEI 1984 ; SLATKIN and BARTON 1989 ; SLATKIN 1991 , SLATKIN 1993 ), and the ratio (51) of expected heterozygosities may also be preferable for data analysis to the expectation of the ratio of random heterozygosities (NEI and CHAKRAVARTI 1977 ; NEI et al. 1977 ). Substituting (47) and (48) into (51) produces the explicit formula

(57)

	DISCUSSION

TOP ABSTRACT DETERMINISTIC GENOTYPIC... STOCHASTIC ALLELIC FREQUENCIES DISCUSSION LITERATURE CITED

Without restricting the evolutionary forces that may be present, we have developed systematically the theory of fixation indices in an arbitrarily subdivided population. Our indices are parameters, rather than random variables. To estimate the pattern and strength of evolutionary forces (such as migration) from the above theory, a model must be specified and used to derive formulas for the fixation indices, as in examples 3 and 4 at the end of this section.

The Equation 26, Equation 28, Equation 31, Equation 33, Equation 51, Equation 53, and Equation 55 for the panmictic indices all have the same simple form: if B is a finer level of subdivision than C, then

(58)

WRIGHT 1969

(59)

We proceed to discuss the interpretation of some measures of population differentiation. According to (10a) and (12a), the fixation index F_ST,ii is a standardized measure of the intersubpopulation variance of the frequency p_i of the allele A_i . By (10b), the corresponding covariance measure for the frequencies of A_i and A_j is F_ST,ij. If every subpopulation is panmictic, then F_IT,ij = F_ST,ij for every i and j, and therefore (5) shows that the parameters F_ST,ij yield the genotypic frequencies in the entire population.

Now consider in more depth the interpretation of the homozygote or heterozygote average index F^*_ST , defined by (32) and evaluated in (33). WRIGHT 1978 (p. 82) noted and exemplified that F^*_ST measures "the amount of differentiation among subpopulations, relative to the limiting amount under complete fixation" and that F^*_ST is "not a measure of degree of differentiation in the sense implied in the extreme case by absence of any common allele. It measures differentiation within the total array in the sense of the extent to which the process of fixation has gone toward completion." These observations suggest that F^*_ST is an appropriate measure of differentiation in a population with low genetic diversity, but that it may be misleading in a highly diverse population. Below, we develop this idea more precisely and illustrate it by four examples.

Since nucleotide diversities are generally low, therefore F^*_ST is usually a suitable measure of differentiation at the nucleotide or codon level.

We separate the cases of high and low genetic diversity and use the criteria of KIMURA and MARUYAMA 1971 ; see also NAGYLAKI 1983 , NAGYLAKI 1985 , NAGYLAKI 1986 .

Our index of genetic diversity is the effective number of alleles (KIMURA and CROW 1964 ; MARUYAMA 1970 )

(60)

NAGYLAKI 1992

For high diversity, our measure of gene-frequency differentiation is . We shall say that differentiation is strong if f_{T S} (defined as 1 ) and weak if f_{T S} (recall that f_{T S} ).

For low diversity, the ratio is insensitive to differentiation because f_{T S} 1. A more sensitive measure is : strong and weak differentiation correspond to _S h_T and _S h_T , respectively.

Now consider

(61)

For low diversity, our criteria are, indeed, equivalent to F^*_ST 1 if differentiation is strong and to F^*_ST 1 if it is weak. For high diversity, however, F^*_{ST S} - f_T , so if f_{T S} 1, then differentiation is strong yet F^*_ST 1; thus, strong differentiation does not imply that F^*_ST 1. Weak differentiation does imply that F^*_ST 1.

Example 1:
Suppose that there are K subpopulations, of which L (0 < L < K) are fixed for A₁ and K - L for A₂ . Then (23c) and (24b) give _S = 0 and h_T > 0, whence (33) yields F^*_ST = 1. This indicates that every subpopulation is fixed, and not all for the same allele. Since there are only two alleles, however, complete differentiation between subpopulations (in the sense of having no common alleles) is possible only for two subpopulations.

Example 2:
By contrast, consider n subpopulations of the same size, without common alleles, each with homozygosity f_S . Then f_T = ¹/_n f_S , so from (33) we obtain

(62)

Thus, F^*_ST < 1 unless f_S = 1, even though the subpopulations are fully differentiated. Furthermore, F^*_ST 1 if f_S 1, whereas F^*_ST 1 if f_S 1. The second possibility is misleading unless carefully interpreted. For high diversity, f_S n (which must always hold if n 1), so F^*_ST 1 for small n, and this result can occur for any n. If diversity is low, then f_S 1 and n must be small, which correctly implies that F^*_ST 1.

Two special cases illustrate the above observations. If n 1, then F^*_ST f_S . If each subpopulation has equally frequent alleles, then f_S = , and hence F^*_ST = .

Example 3:
Our third example is the island model (MORAN 1959 ; MARUYAMA 1970 ; MAYNARD SMITH 1970 ; NAGYLAKI 1983 , NAGYLAKI 1986 , and refs. therein). Generations are discrete and nonoverlapping. Each of n (2) panmictic (including selfing) subpopulations comprises N monoecious, diploid individuals. These colonies exchange gametes with no spatial effect on dispersion, i.e., if the migration rate is m (0 < m < 1), every colony receives a proportion of its gametes from each of the other colonies. Selection is absent, and every allele mutates to new alleles at the same rate u (0 u 1).

We posit that migration is weak and that mutation is weak relative to the stronger one of migration and random drift:

(63)

Then, at equilibrium,

(64)

(NAGYLAKI 1983 ), where N_T = nN represents the total population number;

(65)

NEI 1975

NAGYLAKI 1983

TAKAHATA 1983

CROW and AOKI 1984

TAKAHATA and NEI 1984

COCKERHAM and WEIR 1987

(66a)

(66b)

(NAGYLAKI 1986 ). Using F^*_ST to assess differentiation would replace (66a) and (66b) by 4mN 1 and 4mN 1, respectively, which is correct if and only if 4N_Tu 1. Thus, F^*_ST provides the correct criterion for differentiation if and only if diversity is low (cf. NAGYLAKI 1983 , NAGYLAKI 1986 ).

Example 4:
Our last example is the unbounded, unidimensional stepping-stone model (MALECOT 1949 , MALECOT 1950 , MALECOT 1951 ; KIMURA 1953 ; NAGYLAKI 1989 , and refs. therein). As in the island model, generations are discrete and nonoverlapping; selection is absent; and every allele mutates to new alleles at the same rate u (0 u 1). There are panmictic (including selfing) colonies of N monoecious, diploid individuals at all the integers. These demes exchange gametes at rates that depend on displacement, but not on initial and final positions separately, i.e., dispersion is homogeneous.

Let w denote the separation between the demes from which genes are sampled. We write the variance of the single-generation gametic displacement as ¹/₂² and introduce the scaled, dimensionless separation

(67)

For weak mutation (u 1) and large neighborhood size (N 1), the probability at equilibrium that two distinct genes sampled from demes separated by a distance w (0) are the same allele is adequately approximated by (NAGYLAKI 1989 , and refs. therein)

(68)

(69)

The expected heterozygosity

(70)

Now consider two demes with scaled separation . The effective number of alleles in these two demes is

(71)

For high diversity, we use as a simple index of differentiation between the two demes. Therefore, differentiation is strong if e^- 1 and weak if e^- 1, independent of ß. For low diversity, the measure reveals that differentiation is strong if

(72a)

(72b)

From (61) we obtain

(73)

Again, F^*_ST yields the correct criterion for differentiation if and only if diversity is low.

	ACKNOWLEDGMENTS

I thank BRIAN CHARLESWORTH, JAMES F. CROW, and MAGNUS NORDBORG for useful comments on the manuscript. This work was supported by National Science Foundation grant DEB-9706912.

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

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