The Expected Number of Heterozygous Sites in a Subdivided Population

Corresponding author: Thomas Nagylaki, Department of Ecology and Evolution, University of Chicago, 1101 E. 57th St., Chicago, IL 60637.

Communicating editor: M. SLATKIN

	ABSTRACT

TOP ABSTRACT GAMETIC DISPERSION DIPLOID MIGRATION DISCUSSION LITERATURE CITED

A simple, exact formula is derived for the expected number of heterozygous sites per individual at equilibrium in a subdivided population. The model of infinitely many neutral sites is posited; the linkage map is arbitrary. The monoecious, diploid population is subdivided into a finite number of panmictic colonies that exchange gametes. The backward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). With suitable weighting, the expected number of heterozygous sites is 4N_eu, where N_e denotes the migration effective population number and u designates the total mutation rate per gene (or DNA sequence). For diploid migration, this formula is a good approximation if N_e 1.

ONE of the most important measures of genetic variability at the molecular level is the expected number of heterozygous nucleotide sites per individual, ₀ . For a panmictic population at equilibrium and without selection, KIMURA 1969 showed that in his model of infinitely many sites,

(1)

EWENS 1974

WATTERSON 1975

Since natural populations are frequently subdivided, considerable effort has been devoted to extending (1) to subdivided populations. LI 1976 proved that (1) holds for the island model without recombination if N_e = N_T, the total population number. SLATKIN 1987 generalized LI's result by demonstrating that (1) holds if (i) there is no recombination; (ii) the backward migration matrix M is symmetric and ergodic; (iii) ₀ is calculated by weighting each deme by the reciprocal of the number of individuals in it; and (iv) N_e = nÑ, where n signifies the number of demes and Ñ denotes the harmonic mean of the subpopulation numbers. STROBECK 1987 established (1) with N_e = N_T for weak evolutionary forces and conservative migration. SEE GRIFFITHS 1981 , TAKAHATA 1988 , TAJIMA 1989 , NOTOHARA 1990 , NOTOHARA 1993 , NOTOHARA 1997 , HEY 1991 , NATH and GRIFFITHS 1993 , and HERBOTS 1994 for related studies.

In this note, we prove for gametic dispersion that, for suitably weighted calculation of ₀ , the Equation 1 holds for every linkage map if M is arbitrary, but ergodic and time independent, and if N_e designates the migration effective population number (NAGYLAKI 1980 , NAGYLAKI 1982 , NAGYLAKI 1983 , NAGYLAKI 1994 ). We show also that (1) is a good approximation for diploid migration if N_e 1.

	GAMETIC DISPERSION

TOP ABSTRACT GAMETIC DISPERSION DIPLOID MIGRATION DISCUSSION LITERATURE CITED

Generations are discrete and nonoverlapping; the monoecious, diploid population is subdivided into a finite number of panmictic colonies that exchange gametes in a fixed pattern. We apply the model of infinitely many neutral sites with an arbitrary linkage map to a gene or DNA sequence. Thus, we posit that the mutation rate per site is so low that mutation occurs at each site at most once and then only at monomorphic sites. This approximation requires that the proportion of polymorphic sites be much less than one. Let u denote the total mutation rate per gene.

At the beginning of the life cycle, every one of the N_i adults in deme i produces the same very large number of gametes, which then disperse independently. Complete random union of gametes follows. Therefore, a proportion 1/N_i of the zygotes whose gametes originate in deme i are produced by self-fertilization. Mutation is next, and finally population regulation returns the number of individuals in deme i to N_i. Thus, random genetic drift operates through population regulation.

Before deriving our results, we introduce some essential concepts and parameters.

Let m_ij designate the probability that a gamete in deme i after dispersion was produced in deme j. In the absence of selection, it is reasonable to assume that the backward migration matrix M = (m_ij) is constant (NAGYLAKI 1992 , p. 135). We posit also that M is ergodic, i.e., irreducible and aperiodic (GANTMACHER 1959 , pp. 50, 80, 88). Irreducibility guarantees that the descendants of individuals in each deme are able eventually to reach every other deme. Aperiodicity precludes pathological cyclic behavior. Given irreducibility, the biologically trivial condition that individuals have positive probability of remaining in some deme, i.e., that m_ii > 0 for some i, suffices for aperiodicity (FELLER 1968 , p. 426). Of course, M must be stochastic:

(2)

Let N_T and _i represent the total population number and the proportion of adults in deme i, respectively:

(3a)

(3b)

By the ergodicity of the nonnegative stochastic matrix M, the eigenvalue 1 of M is simple and exceeds all other eigenvalues in absolute value; we may choose the left eigenvector corresponding to this unit eigenvalue to have only positive components (GANTMACHER 1959 , Chapter 13). Thus, the conditions

(4)

The components of the last equation in (4) are

With the aid of (2), we can rewrite this as

Conservative migration patterns are those that do not change the subpopulation numbers; in this case, and only in this case, we have = (NAGYLAKI 1980 ). Conservative migration has many simple intuitive properties that do not always hold for arbitrary migration (NAGYLAKI 1980 , NAGYLAKI 1982 , NAGYLAKI 1983 , NAGYLAKI 1985 , NAGYLAKI 1986 , NAGYLAKI 1992 , pp. 135–136, 151; NORDBORG 1997 ). In our model, the subpopulation numbers N_i refer to adults. However, since the number of gametes in each deme before dispersion is proportional to N_i, it is also true that the gametic numbers are unchanged by conservative migration, and only by conservative migration.

In our results, the vectors and enter combined in the migration effective population number N_e, defined by (NAGYLAKI 1980 , NAGYLAKI 1982 , NAGYLAKI 1983 , NAGYLAKI 1994 )

(5)

We have ß 1 and hence N_e N_T, with equality if and only if migration is conservative (NAGYLAKI 1980 ). This effective population number replaces the actual total population number in the strong-migration limit (NAGYLAKI 1980 , NAGYLAKI 1983 ) and in certain aspects of geographical invariance (NAGYLAKI 1982 , NAGYLAKI 1994 ). Observe that N_e is independent of the genetic model.

There is a simple, intuitive interpretation of N_e (cf. NORDBORG 1997 ). In many cases, 1/N_e can be defined as the probability that two randomly chosen gametes in distinct individuals are descended from the same parent (CROW and DENNISTON 1988 ; CABALLERO and HILL 1992 ; NAGYLAKI 1992 , pp. 243–247, 1995). For gametes in demes i and j, this probability is

Averaging this with respect to the stationary distribution and using (4), (3a), and (5) yield

We are now prepared to deduce our results.

Let T_ij denote the mean coalescence time (in generations) of two distinct, homologous nucleotides chosen at random from adults just before gametogenesis, one from deme i and one from deme j. At equilibrium, considering ancestry and coalescence in the preceding generation yields directly

(6)

Clearly, (6) applies also to a model with 2N_i haploid individuals in deme i.

Define the global and local means (cf. NAGYLAKI 1982 )

(7)

Averaging (6) according to (7) and appealing to (4) and (3a) yield

(8)

Therefore, summing mutations over sites shows that the expected number of nucleotide differences is

(9)

Thus, with the weighting (7), the exact effect of population subdivision in (8) and (9) is to replace the actual total population number N_T by the migration effective population number N_e.

In the strong-migration limit, N_i for every i with and M fixed. Then T_ij ~ 2N_e for every i and j (NOTOHARA 1993 ), where the notation means that T_ij/(2N_e) 1. This demonstrates independently the asymptotic validity of the exact Equation 8 whenever migration dominates random drift.

We discuss special cases of (8) after presenting a different proof of (9).

An alternative proof of (9):
Since (8) and (9) are geographical-invariance relations, the following instructive approach is natural. Suppose the model of infinitely many alleles (MALECOT 1946 , MALECOT 1948 , MALECOT 1951 ; WRIGHT 1948 ; KIMURA and CROW 1964 ) applies to each site: every nucleotide at site s mutates to new nucleotides at rate u_s. (We soon let u_s 0, so the fact that there are really only four nucleotides will not matter.)

Let f^(s)_ij denote the probability that two distinct nucleotides at site s chosen at random from adults just before gametogenesis, one from deme i and one from deme j, are the same. Define the weighted means (NAGYLAKI 1982 )

(10)

(11)

NAGYLAKI 1982

(12)

To obtain the expected number of heterozygous sites in the model of infinitely many sites, we must let u_s 0 and sum over s in (12). Evidently, ^(s) 1 as u_s 0, so (12) reduces to

(13)

(14)

(15)

Conservative migration:
If migration is conservative, then N_e = N_T, so (9) reduces to a result established by STROBECK 1987 for weak evolutionary forces. In this case, = , and hence the averages in (8) and (9) simplify to weighting by the demic proportions:

(16)

Examples of conservative migration are random outbreeding and site homing (NAGYLAKI 1992 , pp. 136, 149, and refs. therein), the island model (NAGYLAKI 1983 , NAGYLAKI 1986 , and refs. therein), and the circular stepping-stone model (NAGYLAKI 1983 , NAGYLAKI 1986 , and refs. therein). The choice m_ij = _j corresponds to panmixia in the entire population.

Note that (16) is independent of the migration pattern, provided the latter is conservative. This raises the following apparent paradox. If there is no migration, then T_ii = 2N_i, whence

(17)

From the formal point of view, note that if there is no migration, then M is the identity matrix. Therefore, contrary to our assumption of ergodicity, M is reducible and is undefined. Thus, (16) does not apply.

A more illuminating explanation is that, as the migration rates tend to zero, so does the probability of descent from a different deme, but the mean interdeme coalescence times (T_ij for i j) diverge and make a finite, positive contribution to the mean intrademe coalescence times T_ii. This behavior is exemplified by the island model with migration rate m: LI's (1976) solution shows that T_ij = O(m^-1) for i j and that the interdeme contribution is O(1) as m 0. For two islands, NATH and GRIFFITHS 1993 demonstrate that the distribution of the intrademe coalescence time converges to the single-deme distribution, but the mean intrademe coalescence time does not converge to the single-deme mean.

Doubly stochastic backward migration matrix:
Here we assume, in addition to (2), that

(18)

(19)

(20a)

(20b)

A natural subclass of doubly stochastic M is homogeneous M: in this case, m_ij = m_i-j, which depends only on displacement, rather than on both the initial and final positions. Examples are the island and circular stepping-stone models, but, as observed above, these migration patterns are also conservative (NAGYLAKI 1992 , p. 136).

Symmetric M is another subclass of doubly stochastic M. In this case, the formula ₀ = 4nÑu was derived by SLATKIN 1987 under the assumption of no recombination. (His mutation rate, however, should be per gene, not per site.)

Two demes:
Parametrizing M as

(21)

(22)

(23)

We can confirm (23) by calculating ₀ from NOTOHARA's (1990, p. 69) formulas for T_ij.

Migration is conservative if = , which is equivalent to N₁m₁ = N₂m₂. This condition means that the same number of individuals migrate from deme 1 to deme 2 as vice versa.

	DIPLOID MIGRATION

TOP ABSTRACT GAMETIC DISPERSION DIPLOID MIGRATION DISCUSSION LITERATURE CITED

In this section, we provide support for the robustness of (8) and (9) by proving that (8) is a good approximation for diploid migration if N_e 1. We derive exact results for conservative migration and weak- and strong-migration approximations for the general case.

We modify the model in the preceding section so that selfing is excluded and zygotes (rather than gametes) disperse, still before population regulation.

Let S_i designate the mean coalescence time of two distinct, homologous nucleotides chosen at random just before gametogenesis from an adult in deme i. Let T_ij signify the mean coalescence time of two homologous nucleotides chosen from distinct adults just before gametogenesis, one from deme i and one from deme j. A moment's reflection shows that at equilibrium,

(24a)

(24b)

We retain (7) and define

(25)

Averaging (24b) and using (4), (3a), (5), and (25), we obtain the invariance formula

(26)

Conservative migration:
Since the number of zygotes in each deme before migration is proportional to the number of adults, therefore it is conservative migration, and only conservative migration, that leaves the zygotic numbers invariant. If migration is conservative, then ₀ and ₀ are averaged with respect to , as in (16). Therefore, averaging (24a) yields

(27)

Recalling that N_e = N_T and solving (26) and (27) simultaneously, we find

(28)

Let ₀ and ₀ denote the expected number of nucleotide differences between two homologous genes in the same individual and in different individuals in the same deme, respectively. Summing mutations over sites, we have

(29a)

(29b)

If N_T 1, then (28) and (29) are very close to (8) and (9), respectively.

One can also deduce (29) by the alternative approach presented in the preceding section: approximate Equation 20aEquation 20b of NAGYLAKI 1985 for weak mutation and sum over sites.

Weak migration:
Let m represent the largest total migration rate:

(30)

We derive an approximation for m 1 and arbitrary subpopulation numbers N_i. From (2) and (30) we see that

(31)

_ij

_ii

_ij

(32)

(33)

The joint solution of (26) and (33) is our weak-migration approximation:

(34a)

(34b)

(35a)

(35b)

Observe that (34) and (35) agree with (28) and (29), respectively, for weak conservative migration.

Strong migration:
If N_i 1 but m / 1, then we can assume that N_i for every i with the backward migration matrix M fixed. In this limit, the result (8) indicates that we must have the asymptotic formulas

(36)

(37)

By the ergodicity of M, the nonnegative, stochastic Kronecker-product matrix M M in (37) has a simple maximal eigenvalue 1, and the corresponding right eigenvector has equal components. Therefore, T^*_ij = T^*, independent of i and j, and hence (24a) implies that S^*_i = T^*. From (26) we get

(38)

This result agrees with (8) and (9), as expected from the strong-migration limit for diploid migration in the model of infinitely many alleles (NAGYLAKI 1983 ).

	DISCUSSION

TOP ABSTRACT GAMETIC DISPERSION DIPLOID MIGRATION DISCUSSION LITERATURE CITED

We have demonstrated that, for gametic dispersion and suitable averaging, the mean intrademe coalescence time is ₀ = 2N_e , and the expected number of heterozygous nucleotide sites is ₀ = 4N_eu , where N_e and u denote the migration effective population number (5) and the mutation rate per gene, respectively. If N_e 1, these formulas are good approximations for diploid migration. Thus, for the simple functionals ₀ and ₀ , population subdivision can be taken into account by replacing the actual total population number N_T by N_e. This reduction generally fails for higher moments of T₀ and d₀ and for measures of genetic variability other than ₀ , such as the expected number of segregating sites.

In contrast, the rate of gene substitution, K, is completely independent of population structure; unlike ₀ and ₀ , it does not depend even on the stationary distribution and the deme proportions . To see this, note first that the fixation probability of a nucleotide with initial frequency p_i in deme i is its weighted average initial frequency,

(39)

(NAGYLAKI 1980 ). For a new mutant that appears in deme j, we have p_j = 1/(2N_j) and p_i = 0 if i j; so P = _j/(2N_j). Since a mutant appears in deme j with probability _j, the unconditional fixation probability is

(40)

KIMURA 1968

(41)

We close this note with some remarks on effective population number. Consult CABALLERO 1994 and NAGYLAKI 1995 for additional discussion and references.

First, it must be kept in mind that the introduction of an effective population number generally does not reduce exactly a complicated model to a simpler or ideal one. At most, such reduction occurs approximately or only for certain functionals of the evolutionary process. For example, the variance effective population number N^(v)_e for a panmictic dioecious population is a parameter (rather than a random variable) only in the diffusion approximation, and it is only in this approximation that it reduces a dioecious model to a monoecious one (EWENS 1979 , pp. 104–112; EWENS 1982 ; NAGYLAKI 1995 ). The inbreeding effective population number N⁽ⁱ⁾_e relates the asymptotic (i.e., long-time) rate of decay of the probabilities of genetic identity (including the heterozygosity) in complicated models to their rate of decay in an ideal population.

Second, although effective population numbers have usually been defined in terms of some property of the evolutionary process, they are theoretically instructive and useful only if they can be evaluated as parameters, rather than random variables that depend on that process. This has been accomplished under a wide range of assumptions for both N^(v)_e and N⁽ⁱ⁾_e .

Third, a particular effective population number is useful only if it can be evaluated without analysis of the evolutionary process or if it predicts more than one property of that process. Again, N^(v)_e and N⁽ⁱ⁾_e satisfy this criterion.

The migration effective population number N_e defined by (5) has all the desirable properties discussed above. Its evaluation from (5) is simple, explicit, and independent of the genetic model: N_e satisfies N_e N_T and depends only on the vector of demic proportions and on the unique stationary distribution of the Markov chain generated by the constant, ergodic backward migration matrix M. Of course, no effective population number can reduce a model of a subdivided population to that of a panmictic one. However, the above and earlier analyses (NAGYLAKI 1980 , NAGYLAKI 1982 , NAGYLAKI 1983 , NAGYLAKI 1994 ) show that N_e replaces N_T in the strong-migration limit and in certain aspects of geographical invariance.

Finally, it should be noted that our definition of N_e differs from that of the various recently introduced effective population numbers for subdivided populations (NEI and TAKAHATA 1993 ; WANG 1997A , WANG 1997B ; WHITLOCK and BARTON 1997 ), which are defined in terms of the behavior of the evolutionary process.

	ACKNOWLEDGMENTS

I am very grateful to MAGNUS NORDBORG for stimulating me to carry out this study by asking whether previous results were special cases of a general formula and for helpful comments on the manuscript. I thank ROBERT GRIFFITHS for a discussion that led to the intuitive interpretation of the migration effective population number. This work was supported by National Science Foundation grant DEB-9706912.

Manuscript received December 5, 1997; Accepted for publication March 16, 1998.

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