AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 77:435-449 (1988) Three Components of Genetic Drift in Subdivided Populations ALAN R. ROGERS Department OfAnthropology, UniuersLty of Utah, Salt Lake City, Utah 84112 KEY WORDS Drift migration, Subdivided populations, Effective population size ABSTRACT Wright’s metaphor of sampling is extended to consider three components of genetic drift: those occurring before, during, and after migration. To the extent that drift at each stage behaves like a n independent random sample, the order of events does not matter. When sampling is not random, the order does matter, and the effect of population size is confounded with that of mobility. The widely cited result that genetic differentiation of local groups depends only on the product of group size and migration rate holds only when nonrandom sampling does not occur prior to migration in the life cycle. The term genetic drif? is used by evolutionists to refer to the effects of a variety of factors that cause stochastic changes in allele frequencies within finite populations. Sewall Wright (1931) is responsible both for the term drift and for the metaphor of sampling that is generally used to discuss it. The statistical effect of drift is likened to that of drawing a sample of gametes, independently and a t random, from the hypothetical infinite population of gametes that the parents could have produced. The ideal of independent random sampling is seldom achieved in natural populations, and departures from it are accommodated by adjusting the effective population size. This simple approach has been enormously successful. By absorbing many complexities into this single parameter, it has allowed geneticists to keep their models simple. The value of this approach is reduced, however, when interest focuses on the complexities themselves. For example, I have been interested in the effects of human patterns of social behavior on the evolutionary forces affecting genetic population differences. It does not seem possible to describe these effects by adjusting the effective population size, or any other single parameter. Accordingly, I introduce here a more detailed model of the action of genetic drift in subdivided populations. The identification of genetic drift with sampling encourages us to regard it as a n event that occurs a t a single point in the life cycle, but this is not the case. For example, a) mated adults are a sample of the reproductively mature adults, b) offspring genes are a sample of the genes of mated adults, c) emigrants from each local group are a sample of (4 1988 ALAN R LISS, INC the individuals born there, and d) reproductively mature adults are a sample of the population at each earlier stage. Each of these episodes of sampling results in genetic drift, but previous models have generally failed to consider them all, or to distinguish between them. A more detailed model will be useful for several reasons. First, it is possible to estimate some of the components of genetic drift even when insufficient data are available for estimation of all components. Second, the components of drift associated with different episodes of sampling are of interest in their own right. We may ask, for example, which episodes of sampling have the largest effects, and whether sampling that occurs before migration has the same effect as sampling after. We may also wonder whether the effects of sampling at different stages interact, enhancing or canceling each other. In previous papers (Rogers and Harpending, 1986; Rogers, 19871, Henry Harpending and I have developed models in which sampling precedes migration. In this work, however, we assumed that effective and actual group sizes were equal. The present paper extends that work by incorporating a detailed treatment of effective population size, and by allowing sampling before, during, and after migration. In the companion paper (Rogers, 1988a), statistical methods are introduced for estimating the parameters of the model from genetic data. Received May 12,1988; revision accepted August 24, 1988. 436 A.R. ROGERS birthplace of each individual. With such data, As Rogers and Harpending (1986) show, all of the quantities above are readily estimost previous theories of evolution in subdi- mated. For convenience, all the notation used vided populations assume a life cycle in in this paper is summarized in Appendix A. which the numbers of individuals migrating The life cycle are very large. The life cycle begins with a n The life cycle assumed in this paper is sumeffectively infinite group of newborns who 1. The first events are remarized in Table then migrate. Following migration, densitydependent mortality reduces each group to production and density-dependent mortality, some fixed number of reproductive adults. which reduce the initial infinite gamete pool Genetic drift occurs just once in this life cycle, of group j to a population of NJ:, newborns. during population regulation, and the effect Next, local migration redistributes these of migration on allele frequencies is deter- newborns among local groups, leaving group j with N+jjuveniles, and then a second round ministic. An alternative life cycle, in which mortal- of mortality occurs, leaving nj adults. The ratio, g = Itj/N+j, is assumed equal in ity precedes migration, seems more relevant to humans and other species in which mor- all groups, and it is further assumed that tality is relatively high among newborns, and migration does not change local population low during the ages of migration and repro- sizes so that N+j = Nj+. Consequently, g is duction. This life cycle has been studied by also equal to nilNi+, the ratio of numbers of Rogers and Harpending (1986) and Rogers parents to numbers of newborn offspring. (1987), who ignored the distinction between Note that g measures the relative imporactual and effective population sizes during tance of early mortality. When there is mortheoretical analysis, but then substituted ef- tality before migration but not after, N + j = fective for actual sizes during data analysis. nj, which makes g = 1. On the other hand, As we shall see, this procedure is unjustified. when mortality does not occur until after The model developed here incorporates a migration, N+j+ 00. For finite nj, this implies more general life cycle that includes mortal- that g+O. Thus, g + l when most mortality ity both before migration and after. The two precedes migration, and g+O when it follows. In the final stage of the life cycle, migrants life cycles discussed above are thus special cases of the one considered here. Before de- are exchanged with a n external “continent” scribing it in detail, it will helpful to define with unchanging allele frequency 7 ~ . This scheme, having two rounds of population resome terms and notation. duction, is similar to one studied by NagyNotation laki (1979). There are three episodes of sampling in Let us refer to the set of individuals that were born in the ith local group and later this life cycle: first, during reproduction and breed in the j t h as the Gth migrant set. The early mortality, second, during local migraiith migrant set thus consists of individuals tion, and third, during late mortality. Deviawho were born in group i and do not emi- tions from the ideal of random sampling grate. When both the birthplace and adult during each episode are measured by three residence of each individual in a genetic sample are known, it is a simple matter to estiTABLE 1: The life cycle mate the allele frequencies of migrant sets. Notation for group j Let qv denote the frequency of allele A1 Group Allele within the i&h migrant set, and N . .the size Stage Process size frequency of this migrant set. The subscript^"+" and a “*’’ will denote summation and averaging, Gamete pool PJ Reproduction respectively. For example, Ni+ = CjNij, the 1 and early size of group i before local migration, qi. = mortality Ni;l Cj Nvqg, the allele frequency in group i Newborn NJ* qJLocal migration before local migration, q.j = N<’CiNgq,j., 1 N+J 4.J the allele frequency in group j after local Juvenile 5 Late mortality migration, and so forth. In surveys of human Adult genetic variation within subdivided popula- 1 External migration tions, it is routine to record not only the n P, residence and genetic phenotype, but also the Adult MODEL GE:NETIC DRIFT IN SUBDIVIDED POPULATIONS parameters, CYE,a ~and , c r ~ each , of which is defined by a n equation of the form Sampling variance = (1 + a) x CRandom sampling variance). Let us now consider each stage in detail. Reproduction and early mortality The gamete pool of each group is a hypothetical infinite sample, drawn a t random with replacement from the genes of the parents. Hence, its allele frequency exactly equals, p,,the allele frequency of the parental population (The superposed tilde here and elsewhere is used to distinguish quantities referring to the parental generation.). Reproduction and early mortality together select a possibly nonrandom sample of these gametes to form the newborn stage of the life cycle. In the ideal case of random sampling, this adds a n increment to the allele frequency with expectation zero and variance pJ1 -pJZN, + (Wright, 1931). In the general case, where sampling may not be random, it is conventional to express this variance as where N$ is the variance effective size (Crow and Denniston, 1988) of oup .i. The difference between Njt. and Nj+. is a measure of the extent to which sampling at this stage of the life cycle is nonrandom. The formulation to be used here is different. Let us rewrite Equation 1 as 437 grandparental as well as the parental generation (Crow and Denniston, 1988). Such formulae are not, to my knowledge, available for subdivided populations with incompletely isolated subdivisions, and would be unwieldy in any event, depending on the migration pattern as well as the mating systems of the various groups. To avoid this complexity, I introduce here a n expression for the inbreeding effective size that involves properties of the parental generation only. There is no magic here-the effect of the grandparental generation is still present. Its effect has simply been absorbed into h, a quantity that is defined below. Unfortunately, the inbreeding effective size has not been defined consistently in the literature of population genetics. Wright (1969, pp. 211-212) defines it in terms of the rate at which heterozygosity decays; Ewens (1979,p. 105)defines it in terms of the probability that two distinct, random genes are copies of the same parental gene; and Crow and Denniston (1988, p. 484) define it in terms of the probability that two uniting gametes are identical by descent a t some locus. Ewens’ usage is followed here. Let us assign to each gene a genic value, which equals unity if the gene is a copy of allele Al, and zero otherwise. Let X j (where j = 1, 2, . . ., 2N,+) denote the genic value of the j t h gene among newborns in group i. Then \ where CYE= N,,lNs - 1. This formulation decomposes the variance due to drift into a “random sampling component,” and a “nonrandom” component that is proportional to (YE, Thus, CYEmeasures the extent to which sampling is nonrandom. Before incorporating this equation into the theoretical argument developed below, let us consider how CYEcan be calculated, and what its value means. It will be helpful to re-express it in terms of the inbreeding effective size of the parental generation (defined below). In populations with sexual reproduction, formulae for the inbreeding effective size generally involve properties of the where Var {xjlp;) = pi(l-pJ, and v is the average correlation between distinct newborn genes, or equivalently, the correlation between two newborn genes drawn a t random without replacement. Define the inbreeding effective size, r& as the reciprocal of the probability that two random newborn genes, drawn without replacement, were derived from the same individual in the previous generation. Two such genes are equally likely to be copies of the same 438 A.R. ROGERS gene or of distinct genes in their common parent. Thus, where fi is the correlation of two genes from the same parent, and ci the correlation of genes from different parents. Since these are correlations of distinct parental genes relative to their own generation, they both equal (2ni - 1)-' at Hardy-Weinberg equilibrium. To re-express c; in terms of fi, note that the correlation between two genes drawn at random with replacement from the parental generation is zero, but can also be written as o=- 2ni Substitution into Equation 3 now leads to Var{qj. Ijjj} = ignoring terms of order (n{ni)-'. The inbreeding and variance effective sizes are thus related by in all groups. Comparing Equation 6 with Equations 1and 2 shows that (YE = q / g . These formulae are unusual in that they express the inbreeding and variance effective sizes of a sexual population without explicit reference to the grandparental generation. This is possible only because the effect of inbreeding in that generation has been absorbed into the parameter fi. The advantage of this approach is that it avoids the complexity introduced by immigration from other local groups. The disadvantage is that the demographic parameters of the population are not sufficient to specify the sampling variance a t this stage of the life cycle-an estimate of fi is also needed. I take fi as a given and will not try to evaluate it here. To clarify the meaning of these formulae, let us apply them to a hypothetical sexually reproducing population in which each local group contains 5 male parents and 45 female parents, who mate a t random to produce 100 offspring. The reciprocal of the inbreeding effective size is the probability that two offspring genes drawn at random without replacement are derived from the same parent. The two genes have probability 1/(2N;+ - 1) = 1/199 of coming from the same offspring, in which case they cannot have come from the same parent. With probability 1 - 1/199, they are from different offspring. Given that they are from different offspring, they are both from female parents with conditional probability 1/4, and both from the same female parent with conditional probability 1/(4 x 45). Similarly, the conditional probability that they are both from the same male parent is 1/(4 x 51, since there are five male parents. Thus, the probability that both genes derive from the same parent is 1 1 --,=-xo ni 199 to a close approximation. Let a~ = (1 fi) (niln; - 1).Then + = where a' depends both on non-random sampling during reproduction and early mortality, and on departures from Hardy-Weinberg equilibrium in the parental generation. In general, at may vary among the local groups, but is assumed here to have the same value 0.0553, and the inbreeding effective size is nt = 110.0553 = 18.091. a1 is now calculated as (1 + fi)(nJnf - 1).Let us assume that the parents are in Hardy-Weinberg equilibrium, so that fi = -1/(2ni - 1)= -1199. This gives a1 = (1- 1/99) x (50/18.091 - 1)= 1.746. In general, the quantity 1 + a1 is approxi- GENETIC DRIFT IN SUBDIVIDED POPULATIONS mately equal to the ratio of the number of parents in a local group to its inbreeding effective size. In human populations, this is often between two and five (Wood, 19871, so CYIshould often fall between unity and four. Local migration The pattern of migration among K local groups is described by the migration matrix, M, whose ijth element is mu = N,/N+,, the fraction of group j derived from group i each generation. I assume that no subset of groups is completely isolated from the others, and that there are a t least a few individuals that do not migrate in each local group. These assumptions guarantee that M will have exactly one eigenvalue equal to unity, and that the rest will be smaller in absolute value. The eigenvalues of M are denoted by A,, and are indexed such that 1 = XI 2 A, > . . . 2 . ,A Since each migrant set is a sample of the offspring born in that local group, drift also occurs during local migration and adds to each allele frequency a n increment with expectation zero. The variance of this increment, and also its covariance with increments from different groups, will depend on the statistical properties of the migration pattern. In particular, if biological relatives tend to migrate together, migration is said to be kin-structured (Fix, 19781, and the sampling variance due to migration is increased. To measure this effect, a model of kin structure is required; the one employed here is based on that of Rogers (1987). The simplest form of kin-structured migration (KSM) is that in which groups of individuals migrate as units, which are all of one size, say y. These groups will often be families of some sort, and Rogers (1987) used the term family to refer to them. This has led to some confusion, since these groups need not be families in the usual sense of the word. Indeed, the theory also holds if these groups are less related than random individuals. Here, I use the more general term rnig.de, which is defined by most dictionaries of natural history as a “unit of migration.” In the present context, it refers to a group of individuals that migrate together, and independently of other migrules. Each migrant set contains one or more migrules, and each migrule contains one or more individuals. The mode1 used here is not a n accurate descrip- 439 tion of any real population because of the unrealistic assumption that all migrules are of the same size. Nonetheless, it will be useful a s a baseline against which real populations can be compared. The task now before us is to find a measure of the non-randomness of sampling during migration. Ideally, this measure should be independent of non-random sampling a t other points in the life cycle, yet neither of the existing formulations of kin-structured migration (Fix, 1978; Rogers, 1987) achieves this. To appreciate the problem, consider the important special case of KSM in which siblings migrate together. In this case, each migrule is the progeny of a single pair of parents. Variation in progeny size will then affect population differences in two distinct ways. First, it affects the effective population size, which influences the variance of qi. about pi (Crow and Denniston, 1988).Second, it affects the variance of migrules about qi.. Only the second of these effects is attributable to KSM, for the first would exist even if migration were independent and a t random. Rogers and Harpending (1986) and Rogers (1987) sidestep this issue by assuming it away-they assume effective and actual population sizes to be equal-thus precluding any consideration of non-random sampling during reproduction and early mortality. Fix (1978) also avoids the issue. He varies the strength of kin structure by adjusting the the variance of migrule (i.e., progeny) allele frequencies about the parental allele frequency. Since each of his migrules is uncorrelated with the others, this adjustment changes the variance effective size ( N s ) of the cohort of newborns as well a s the strength of kin structure. The formulation introduced here makes it possible to separate these effects. Consider the statistical model where .xirs is the allele frequency (0, 112, or 1) of the sth individual in the rth migrule born in group i, and where yir and tirs are migrule and individual effects, respectively. Since the numbers of individuals per migrule, and of mimules Der birthdace are small, the varianie components associated with this model must be derived using theory appropriate for 440 A.R. ROGERS finite populations. The terms of this model can be defined so that c tirs = 0, for all i and r, and ($1 (1 - Y - 1 / Cyi, = 0 , for all i. Noting that the number of migrules from D O U P i is N i + h , we can write the mean squares within and between the migrules of the ith birthplace as respectively. Ultimately, we will be interested in the variances and covariances of the allele frequencies of local groups. These can be expressed in terms of the expected squares and products of allele frequencies of individuals. There are three cases to consider-the square of a n individual’s allele frequency, the product between the allele frequencies of migrule members, and the analogous product for individuals from different migrules. Searle and Fawcett’s (1970)results can be used to justify the following expressions: where r f r’ and s z s’. Some way is needed of summarizing the extent to which migration is kin-structured. Rogers (1987) proposed a measure, 0, that is not easy to estimate with genetic data. Here, I introduce another measure that is nearly identical in value, but much easier to mani = uty’ + u, ulate and estimate. Let ut” denote the variance of individual allele frequencies about qi.. If migrules were formed by sampling a t random with replacement, the variance of their allele frequencies would be uin/y, since each is of size y. Under sampling without replacement, this variance would become by analogy with the variance of the hypergeometric distribution. Thus, we can define , the extent to which sama measure, O ~ Mof pling during migration is non-random by In a n earlier paper (Rogers, 1987), I showed , y is the size of each that 0 = (y - l ) ~where migrule and K the correlation within migrules. This approximation was then used to estimate 0 from behavioral data. The same approximation can also be justified for the new measure, (YM. Equation 10 implies that This correlation is relative to the current generation, rather than to the infinite founding stock referred to by the classical formulae of Wright (1969) and Malecot (1969). However, it can be shown that these formulae are approximately correct for first degree relatives if Ni+ > 100, and for second-degree relatives if N i , > 1000 (Rogers, 1986). Thus, if full sibs migrate together, K = 0.5 in all but the smallest populations. By rearranging Equation 13, it can be shown that PF; ignoring terms of order y/Ni+. Thus, if migrules are small compared to local groups, the product (y - 1 ) provides ~ a useful approximation of C U M .This relation also holds approximately for 0 (Rogers, 19871, which shows GENETIC DRIFT IN SUBDIVIDED POPULATIONS that o l and ~ 0 are nearly equivalent. This relation also allows D M to be estimated from information about migrule size and the correlations within migrules. If y and K vary little between local groups, CYM will also be invariant or nearly so. For simplicity, I will assume that CYM has the same value in all groups. Late mortality The final component of drift occurs during late mortality, which removes a portion of the population prior to breeding. This process adds a n increment, qj, with mean zero and with a variance that depends on the extent to which sampling is nonrandom. Under random sampling, the variance is 441 The equation describing evolutionary change Putting all this together, we have p = (l-s)M*P + ST^ + E, (15) where p is a column K-vector whose ith element is pi; 1, a column K-vector each element of which is unity; 6 , a column K-vector of deviations due to genetic drift. Equation 15 differs from Rogers and Harpending’s (1986) Equation 15 only in the definition of t . For the present model, the j t h entry of t is = (1 - ~ ) ( q . j - E(q.jlP) + $ (16) Equations 15 and 16 summarize the evolutionary dynamics of the model, and will be used to predict the equilibriumvalues of various measures of genetic variation. MEASURING VARIATION AMONG GROUPS by analogy with the variance of the hypergeometric distribution. Thus, we can define a measure of olL of non-random sampling during late mortality by Genetic variation among groups can be measured in a variety of ways. The measure adopted here is a variant of Wright’s F-statistics, and is defined in terms of the K x K genetic correlation matrix, K,whose ijth entry is Let w,= N + ,JN++ denote the relative size of the ith group, and W a diagonal matrix whose ith diagonal entry is w,.A useful summary measure of genetic differentiation is The covariances,E{qjqk), are zero f o r j # h. K ro = c w, rtr = tr{WR}, 2=1 Systematic pressure I assume that some form of systematic pressure-mutation, migration, or weak selection-tends to move each local population toward some intermediate allele frequency. The strength of this force is measured by the parameter s, which, for concreteness, is here taken to be the fraction of each local group replaced per generation by emigrants from an external continent with unchanging allele frequency K. For simplicity, external migration is treated deterministically, although local migration is stochastic. This simplification has little effect on the answers provided that external migration is relatively weak. (17) where td.) denotes the trace (sum of the diagonal elements) of a matrix. The expectation of ro is denoted by p = E(r0). Note that ro is defined in terms of variation about the mean of the current generation, p.. My ro is equivalent to one of the senses of Wright’s (1951)Fst. See Rogers and Harpending (1986) for further discussion of the relationship between these notations. These quantities are defined in terms of the allele frequencies of adults. We can also define, for newborns, 442 A.R. ROGERS R‘ = [ r ’ ~ ]f, 0 , = tr(WR’), and p’ = E { r ’ o } , where W’ is the diagonal matrix of relative group sizes prior to migration. The results presented below assume that migration does not change the sizes of local groups, so that W’ = W. As Rogers and Harpending (1986) have emphasized, r-statistics for newborns and adults can can differ greatly in magnitude. It is often convenient to work with the quantity G=-, P 1 - P G can be interpreted as the ratio of expected genic variance between groups to that within, and is a parameter of considerable biological interest. For example, it can be shown that the effectiveness of Wright’s (1945) mechanism of intergroup selection is proportional to G (Rogers, 1988b). Rogers and Harpending (1986) show that Equation 15 implies that, a t equilibrium between migration and genetic drift, m Here Lt is the tth power of the “reduced migration matrix”, and is defined by L = Mt(I - w l q , where I is the identity matrix. w is a column vector whose ith entry is wL, and C = E { E E ~ } / @ -p.)) . ( ~ is a matrix of normalized variances and covariances due to one generation of genetic drift. To find the expectation of R under the current model, we use Equation 16 to find C and then substitute into Equation 18. This calculation is done in Appendix B. RESULTS In the appendix are derived approximations for expectations at equilibrium of the R-matrices of both adults and children, and also for p and p ’ . It is shown that, for small s, Non-random sampling c Random sampling Reproduction andearly mortality DISCUSSION h c I - - --- G = (1-s)’ Migration Late mortality + (I - ~rn,)a,+ Znt,gaM + (I -. gX1 where m, is the effective migration rate, defined by Rogers and Harpending (1986) as one half the harmonic mean of ( 1 - (1 syh? : i = 2,3, . . .,K } . The first term in the numerator refers to the effect of random sampling, the others to nonrandom sampling. When sampling is random, these latter terms vanish. The effect of nonrandom sampling is subdivided into separate effects for reproduction and early mortality, migration, and late mortality. The result of Rogers (1987) is a special case of Equation 19 obtained by setting g = 1, and cq = CUL= 0. Rogers and Harpending’s (1986)results are obtained by adding the additional restriction that c4!M = 0. Much has been made of the observation that, in most models of migration and genetic drift, G depends on migration rate and effective group size (say N,) only through their product N,m, (Lewontin, 1974, p. 213). Equation 19 shows that this result holds only if the second and third terms in its numerator are nil, i.e., if a1 = 0 and either g = 0 or CYM = 0, which requires that any sampling occurring prior to migration be random. In that case, Equation 19 could be reduced to the traditional form, G = l/(N,m,), by a suitable redefinition of N , and m,. When nonrandom sampling occurs early, however, no such simplification is possible. We might define N , as the ratio of n+K/(K - 1)to the numerator of Equation 19. This, however, would make N , a function of the migration rate, so it would make little sense to claim that G depended only on the product of group size and migration rate. Thus, it seems best to leave Equation 19 as it stands. Any attempt to collapse the numerator into a single parameter would obscure the complexity of the relationship between group size and migration rates in subdivided populations. The effects of nonrandom sampling depend both on when in the life cycle it occurs, and on the level of mobility among populations. These dependencies are summarized in Table 2, discussed in detail below. -sPaL In+m&l(K - 1) (19) In general, the ratio, G, of expected between- to expected within-group variance is influenced by three episodes of genetic drift involving: a) reproduction and early mortality, b) migration, and c) mortality after migration. All three of these episodes can be interpreted using Wright’s metaphor of sampling: newborns constitute a sample of the 443 GENETIC DRIFT IN SUBDIVIDED POPULATIONS TABLE 2. Parameters with dominant influence on G as determined by mobility and the timing of mortality Mobility Mortality Early ( g = 1) Late (g = 0) Low (me = 0 ) High (me = 0.5) “t “M a17 “ L Q,L hypothetical infinite ensemble of newborns that their parents could have produced, migrants a sample of the individuals born in their local group, and reproductive adults a sample of the local group just after external migration. Random sampling In the simplest case, when sampling is at random in all three stages, the order of events does not matter. This can be seen by setting a1 = CUM = c r ~= 0, in Equation 19, and noting that G is then independent of g, which measures the relative importance of early (premigratory) mortality. This implies that sampling prior to migration is equivalent to sampling during or after migration. It was this equivalence that led Harpending and me to conclude (Rogers and Harpending, 1986) that the details of the life cycle had little effect on the variance among groups of adult allele frequencies. As is now clear, that conclusion was a n artifact of our assumption of random sampling. Nonrandom sampling In real populations, the assumption of random sampling may fail, and the order of events becomes important. Furthermore, the effect of drift is confounded with that of mobility. Four extreme cases may be distinguished, as shown in Table 2. Reproduction and early mortality. The parameter QI measures the departure from random sampling during reproduction and early mortality, and is also influenced by departures from Hardy-Weinberg equilibrium in the parental generation. Variation among adults in reproductive success, and unequal numbers of male and female parents, for example, both affect G through the parameter q. The quantity 1 + is approximately equal to the ratio of the size of the adult population to its inbreeding effective size. Thus, Q I can be estimated using the wellknown formulae for effective popGIation size (Crow and Denniston, 1988). In human populations, estimates of effective population size are usually between 1/5 and 112 the actual size (Nei, 1969; Wood, 1987). These values correspond to values of q between unity and four. As Table 2 indicates, this effect is only important when mobility is low. In some populations, group differences may be nearly independent of the inbreeding effective group size. Migration. Nonrandom sampling during migration may inflate (or deflate) the variances of migrant sets about the allele frequency of the natal group, a n effect that is measured by the parameter a ~The . form of nonrandom sampling considered here is kinstructured migration (Fix, 1978), which occurs when relatives migrate together. As Table 2 indicates, this effect is most important when mortality is early and mobility high. Levin (1988) and Levin and Fix (in press) suggest that kin-stuctured migration may have a n important influence on the genetic structure of plant populations, since some plant migrules are fruits containing many seeds. In some cases, however, the fraction of plant migrules surviving late mortality ( g ) may be low. If so, the present model suggests that kin-structured migration may have little direct effect. I have shown previously (Rogers, 1987)how behavioral data can be used to approximate 0, which is approximately equal to o l ~Those . results indicate that CXM = 0.53 for male lions, and that aM = 3 if entire human sibships dispersed as units. These numbers indicate that the potential effect of kin-structured migration is about a s large as that of reproduction and early mortality. The companion paper (Rogers, 1988a)develops statistical methods for estimating a M from genetic data,Application of this method to data from the Aland Islands, Finland, shows that c x ~ is near zero there. The values of CYM and a)[ may often be related. For example, in Fix’s (1978)model (discussed above) the variance among migrules is inversely proportional to a parameter A, and there is no mortality after migration. In terms of the model introduced here, Fix’s assumptions imply that g = 1,that 1 + a1 = A -I, and that 1 - 1l2N l+Ct,YM= A - 1f2N = 1+ CYI, 444 A.R. ROGERS where N is the size of local groups. Thus, the form of kin structure envisioned by Fix has equal effects on the first two of the three episodes of sampling studied here. Lute mortality. Finally, nonrandom sampling during postmigratory mortality is measured by aL, and is important when mortality is late, regardless of the level of mobility. To my knowledge, this component of genetic drift has never been measured, and its magnitude in natural populations is a mystery. It is easy, however, to envision mechanisms that could induce substantial values of a ~ . For example, intense competition within migrules might generate a negative correlation between the survivorship of' migrule members, leading to a negative value of aL. On the other hand, suppose that postmigratory mortality were entirely due to some fatal infectious disease, and that the rate of transmission within migrules were much higher than that between. This would induce a positive correlation between migrule members that would make a~ a function of a ~In. fact, if migrules survive or perish as units, it can be shown that q = a ~This . shows that aL is potentially as large as aM-values in the range from unity to three are reasonable for all three as. Male lions present an example of a case in which the survivorship of migrule members is likely to be positively correlated. Before breeding, they must leave the pride in which they were born and drive out the resident male in some other pride. Lions that, disperse in groups seem to have a better chance of doing this than those that disperse individually (Bygott et al., 1979). Another example is found in Levin and Fix's suggestion that kin-structured migration may be important in plants if the migrule is a fruit containing several seeds. Levin (personal conimunication) has observed that survivorship of plant migrule members may also be positively correlated in some cases. The fate of all the seeds in a fruit, for example, may depend largely on whether the fruit lands on a favorable or an unfavorable patch of habitat. On the other hand, these seeds may also compete with each other, inducing a negative correlation between the survivorship of migrule members. The sign and magnitude of these correlations are empirical questions whose answers are as yet unknown. CONCLUSIONS In subdivided populations, the effect of genetic drift is more complicated than most previous models have indicated. It depends both on the timing of events in the life cycle, and also on the level of mobility among local groups. At each stage in the life cycle, the sampling variance due to drift can be partitioned into a random sampling component and a nonrandom component. The ratio, G, of between- to within-group genetic variance is insensitive to the timing of the random component, but sensitive to that of the nonrandom component. Mobility reduces the importance of nonrandom sampling occurring prior to migration, enhances the importance of that occurring during migration, and has no effect on that occurring after. The component of genetic drift produced by nonrandom sampling during migration is potentially about as large as that due to reproduction and early mortality. The magnitude of the component due to nonrandom sampling during postmigratory mortality has never been evaluated, but there is no obvious reason to assume it to be negligible. Thus, all three components may be important. Most previous studies of the influence of social and demographic factors on genetic drift have concentrated on changes in the effective population size. Yet this cannot account for all three of the episodes of sampling that occur in subdivided populations. A comprehensive study of drift in a subdivided population would compare the statistical properties of sampling before, during, and after migration. ACKNOWLEDGMENTS I thank J. Boster, E.A. Cashdan, J.F. Crow, D. O'Rourke, and J.W. Wood for comments on this paper. Supported in part by NIH grant MGN 1 R29 GM39593-01. LITERATURE CITED Bygott GD, Bertram BCR, and Hanby JF' (1979) Male lions in large coalitions gain reproductive advantages. Nature 282839-841. Crow JF and Denniston C (1988) Inbreeding and variance effective population numbers. Evolution 42:482495. Ewens WJ (1979) Mathematical Population Genetics. New York: Springer Verlag. Fix A (1978) The role of kin-structured migration in genetic microdifferentiation. Ann. Hum. Genet. 41:329-339. Harpending HC, and Ward RH (1984) Chemical systematics and human populations. In MH Nitecki (ed.): Biochemical Aspects of Evolutionary Biology. Chicago: The University of Chicago Press, pp. 213-256. Levin DA (1988)Stochastic elements in plant migration. Am. Nat. (in press). Levin DA, and Fix AG (1988) A model of kin migration in plants. Theoretical and Applied Genetics (in press). GENETIC DRIFT IN SUBDIVIDED POPULATIONS Lewontin RC (1974) The Genetic Basis of Evolutionary Change. New York: Columbia University Press. Malecot G (1969) The Mathematics of Heredity. San Francisco: Freeman. Malecot G (1973) Isolation by distance. In NE Morton (ed.) Genetic Structure of Populations. Honolulu: University of Hawaii Press, pp 72-75. Nagylaki T (1979) The island model with stochastic migration. Genetics 92:163-176. Nei M (1969) Effective size of human populations. Am. J. Hum. Genet. 22694-696. Rogers AR (1986) Correlations between relatives in small populations. Am. J. Phys. Anthropol. 7Zr377-389. Rogers, AR (1987) A model of kin-structured migration. Evolution 41:417-426. Rogers AR (1988a) Statistical analysis of the migration component of genetic drift. Am. J. Phys. Anthropol. 77,451-457. Rogers AR (1988b) Group selection by selective emigration: The effects of migration and kin structure. Am. Nat. (submitted). Rogers AR, and Harpending HC (1986) Migration and genetic drift in human populations. Evolution 40:13121327. Searle SR, and Fawcett RF (1970)Expected mean squares in variance components models having finite popula!.ions. Biometrics 26:243-254. Strang G (1976) Linear Algebra and Its Applications. New York: Academic Press. Wood JW (1987)The genetic demography of the Gainj of Papua New Guinea. 2. Determinants of effective population size. Am. Nat. 229r165-187. Wright S (1931) Evolution in Mendelian populations. Genetics 26:97-159. Wright S (1945)Tempo and mode in evolution, a critical review. Ecology 26r415-419. Wright S (1951) The genetical structure of populations. Ann. Eugen. Z5:323-354. Wright S (1969) Evolution and the Genetics of Populations. Volume 2. The Theory of Gene Frequencies. Chicago: University of Chicago Press. APPENDIX A: NOTATION This appendix summarizes all notation appearing above, but not that in the appendices below. Primes (as in r’) are used to distinguish quantities referring to newborns from those referring to adults, and tildes (as in pi) to distinguish quantities referring to the previous rather than the present generation. 1 a column vector each element of which is unity; aE = N,+/Ny+ = qlg, a measure of nonrandom sampling during reproduction and early mortality; a,r= (1 + f i ~ n ~ n fI), an alternate measure of nonrandom sampling during reproduction; aL,= a measure of nonrandom sampling during late mortality; aM = a measure of nonrandom sampling during migration; t = a vector of deviations due to genetic drift; y = the size of each migrule; K = the genetlc correlation of individuals within migrules; A, = the ith eigenvalue of the migration matrix, where 1 = X1 > Xz >. ’ . 2 h ~ ; q = vector of increments t o allele frequencies produced by late mortality; = P = Ejro}; = the allele frequency 445 of the external continent, which is assumed not to change; tLrS = deviation of the sth individual in the rth migrule from his migrule allele frequency; ci = correlation between two alleles, drawn at random from distinct individuals of the parental generation of group i; C = E{ft*}/(p.(l - p.)), a matrix of normalized variances and covariances due to one generation of genetic drift; fi = correlation of two genes from the same parent relative to the parental generation; g = n J N , + , the ratio of number of parents to number of newborns in group i; because of the assumption that N,+ = N + i ,g is also the fraction of group i that survives late population regulation; G = p/(l - p), the ratio of expected variance between groups to that within; I : the identity matrix; K = the number of groups; Lt = Mt(I - wl‘), the tth power of the “reduced migration matrix”; rnd = NLJIN+J7 the fraction of group j derived from group 1. each generation; M = Lm,],. the migration matrix; rzj the size of group j after late mortality; n, - the inbreeding effective size of group i; NV = the size of the ijth migrant set; N,+ = FINc,,the size of group i before local migration; N , , = C,N,,,the size of groupj after local migration; N!: . . = the variance effective size of group i ; pi = the frequency of allele A l at the reproductive stage in group j ; 9.] = N;,’C,N,J9,J, the allele frequency in group j after local migration; y , . = N;tC,N,q,, the allele frequency in group i before local migration; q g = the frequency of allele A l within the ijth migrant set; ril = (p, - p.)l(p, - p.Y(p.(l - p.)), the genetic correlation between groups i and j ; K = [rtJ],the genetic correlation matrix; ro = CLrLiwL, a measure of group differentiation that is equivalent to one of the senses of Wright’s Fst; s = the strength of systematic pressuremutation, migration, or weak selectiontending to move each local population toward some intermediate allele frequency (for concreteness, s is here taken to be the fraction of each local group replaced per generation by emigrants from a n external continent); vf’ = mean square of individual effects within group 2; v(-v’= mean square of migrule effects within group i; w,= N,+IN+ +, the relative size of the ith local group; w = a column vector of relative population sizes; W = a diagonal matrix of relative population sizes; xJ = the genic value (0 or 1)of the j t h gene among newborns in group i; nlrp= allele frequency (0, %, or 1)of the sth individual in the rth migrule in the ith local group; and ycr = deviation of the rth migrule in the ith local group from the allele frequency of that local group. ?r 7 446 A.R. ROGERS APPENDIX B: DERIVATION OF CENTRAL RESULTS This appendix derives formulae describing the amount and pattern of genetic variance expected among a set of K local groups at equilibrium between the effects of migration and genetic drift. Notation and assumptions are given above. To find the equilibrium values of E { R } and p , we must find C under the present model and substitute into Equation 18. By definition, C = E { E E ~ ) / ( P-, (p~J . In view of Equation 16, this is C = (1 - d2 (Z p.(l - p.) + Y), members, and NG(NG - y) terms for members of different migrules. Conditioning on qi. and substituting Equations 9, 21, and 22 into 23 produces where (20) I -1, otherwise. where Y = [E{qiqj}] is a matrix of variances and covariances due to the late morThe matrix Z tality component of genetic drift, and Z = Z contains the expected squares and cross[El Cou{q. ',q.k ( p } ]is the corresponding matrix for the early mortality and migration products of components. Expected squares and products of allele frequencies of individuals and migrant sets Calculation of Z will require expressions The j k t h entry of Z is thus for the squares and products of allele freK quencies of individuals and migrant sets. These are derived from Equations 9, 10, and zjk = mijmikCou{qij,qik(fii}, (25) i= 1 11. The latter two equations can be rewritten, using Equation 12, as where cou{ qLJ,qikI pi} is the conditional covariance of mimant set allele freauencies given the parencal allele frequency. &me the effects of drift in different groups are un, (21) correlated, terms such as cov{q,j.qi,k}(where i # i') are zero, and do not appear in Equation 25. - vim (1 + 'YM) The conditional covariances above can be . (22) decomposed into components due to early Ni+ - 1 mortality and migration: We will also need the expectations of products of the form qGqik. Each of these is itself a sum of products: where ydu is the allele frequency (O,?h, or 1) of the uth individual in the ijth migrant set. I f j # k , all the terms in this sum are for individuals of different migrules, and their expectation is given by Equation 22 On the other hand, if j = k , there are NG squared terms, NG(y - 1)terms for pairs of migrule The term on the far right is the early mortality component of genetic drift, and is given by Equation 6. The other term is the migration component of drift, and is obtained from Equation 24 by taking the conditional expectation of vtn, given pi. If mating is random within groups, then u p = qi.(l - qi.)/2, and its conditional expectation is 447 GENETIC DRIFT IN SUBDIVIDED POPULATIONS Substituting into Equation 26 yields The matrix Y Y contains the variances and covariances produced by late mortality, i.e., Yik = E{yjvk}. Since mortality affects allele frequencies in different groups independently, the off-diagonal entries of Y are zero. The diagonal entries are given by Equation 14, which can be written as Here I introduce the approximation pi(1 pi) = p.(l - p.)(l - ro), i = 1,...5.This approximation has often been used in similar models (Malecot, 1973; Harpending and Ward, 1984) and its accuracy has been verified by Rogers and Harpending (1986). With this approximation, Equation 25 becomes = IjJl - p.X1 - For simplicity, I will a) take p.(l - p.) (1 ro) as a crude approximation for q.j(l - qOj), and b) assume that late mortality does not change relative group sizes so that njln. = N,j/N, + = wj.These simplifications provide Y = p.(l - p.1cw-1, (30) where ro) The matrix C where O(N<%) contains high-order terms, and Substituting Equations 28 and 30 into will be ignored. Equation 20, and then Equation 20 into 18 F o r j = k , this is equal to produces =: JJ @.(l- p.x1 - ro) E{R) 2N++ K 1=1 = (1 - d2~ L ~ R + (L b + dx), (31) where w, where wJ = N+jlN++is the relative size of the ith local group. F o r j # 12, x These last two formulae can be written in where is a diagonal matrix containingihe eigenvalues of L, which are denoted by A;, i matrix notations as = 1,2,...,K. Theorem 4 (Appendix C) shows (28) that these eigenvalueg are identical with z -- p.(i - p.) ( a ~ W -+ lhw--I), ~ those of M except that hl = 0, whereas A1 = 1. The second line above is justified by Theowhere 448 A.R. ROGERS rem 5 (Appendix C ) . Using the formula for the distinction between p . and p.. The last the sum of a geometric series, this becomes line is justified by Theorem 3 in Appendix C. Since ''0 = tr{WR'}(see Eq. 171, its expecX = VDVT, (32) tation is where D is a diagonal mairix whose ith di19 + crz)(l - PI agonal element is zero if X i = 0, and (1 p ' = p + tr{W (1 - sf%&' otherwise. 2n + The expression that results from substitut( g + az)(1 - P ) ing Equation 32 into Equation 31 can be = p + , (35) 2; simplified using the fact that V% = xVT, which holds because V T contains the left eiwhere fi = n/(K - 1). The second line is genvectors of L. This yields justified by part e of Theorem 3 (see AppenE{R} = VBVT, (33) dix C). Rearranging this expression leads to where B is a diagonal matrix whose_ith diagonal entry is zero if i = 1(i.e., if X i = O), and B,, (1 = px1 - SY 1 - (1 - s)2h,2 .( 1+ QM + (aiE - a.w)hL2 (1 + 2N++ - (1 - px1 + ar,x1- - SY 2n+ g) (34) APPENDIX C: MATRIX THEOREMS This appendix derives several results concerning the matrices discussed above. Most + @M)1 - (1 - s y h f of them were first derived (to my knowledge) by Harpending and Ward (1984) or Rogers The approximation here is crude unless s 4 and Harpending (1986). Xz. Rogers and Harpending also show that p Let A = [Nij/N++]denote a matrix whose is equal to the sum of the diagonal entries of yth entry is the fraction of the total populaB. Summing Equation 34 over i and dividing tion that moves from group i to group j each by 1 - p leads to Equation 19. generation. The migration matrix is related to A by M = AW-I, where W is a diagonal matrix of relative population sizes. 2n+ . ( g + ark? + (1 - gx1 + ar,) Newborns The results just derived refer to allele freTheorem 1 quencies of "adults," i.e., of individuals after If the migration pattern is symmetric, i.e., migration and late mortality. This section if A = AT, then extends those results to cover allele frequencies of "newborns," i.e., of individuals after a. The migration matrix can be written in early mortality but prior to migration. Let diagonal form as M =UAVT, where the colR' denote the matrix of normalized genetic umns of U and V contain, respectively, the covariances among the group allele frequen- right and left eigenvectors of M, and where cies of newborns. Rogers and Harpending A is a diagonal matrix containing the eigen(1986)show that values of M; b. U = W%S, where S is an orthogonal E{R'} = matrix, i.e., where SST = I. c. v = w-%. - E{Rj Proof Since A is symmetric, so is the matrix = E ( R } + (g + ad(1 - P ) W-'H. X = WPMAW-",and the spectral theorem 2n+ (Strang, 1976) ensures that it has a diagonal form X = SAST, where S is orthogonal and A where D i u g { . ) is a diagonal matrix whose diagonal. Now M = W%XW-", which is jth diagonal entry is the quantity within equal to UAV', as claimed. Since U is the braces. This expression uses the same ap- inverse of V T , and A is diagonal, this must proximation as Equation 25 and also ignores be the diagonal form of M. + (g + ad(' 2n+ - P) Hw-IH, GENETIC DRIFT IN SUBDIVIDED POPULATIONS 449 Theorem 2 Let 1 denote a column K-vector with each entry equal to unity. If the migration matrix is symmetric so that A = AT, then uj, respectively, the Jth row and column vectors of M. Since M is a stochastic matrix, its leading eigenvalue is unity. If this eigenvalue is unique (i.e., has multiplicity one), then a. A1 = AT1 = w; i.e., relative population sizes may be obtained by summing either the rows or the columns of A. b. Mw = w; i.e., w is a right eigenvector of M associated with a n eigenvalue of unity. c. 1% = lT;i.e., lTis the corresponding left eigenvector of M. a. Liw = 0; i.e., w is a right eigenvector of L' with eigenvalue zero; b. 1TL' = 0; i.e., lTis a left eigenvector of L' with eigenvalue zero: c. For j > 1, L'uj = M'uj = Xjiuj.In other Proof a, The sum of the rows of A are the relative population sizes before migration, while the row sums of AT give the relative sizes after migration. The symmetry assumption ensures that these sums are the same. b. Rewriting M as AW-' transforms proposition b into AW-lw = A1 = w, because of proposition a. c. Making the same substitution in proposition c gives lTAW-l = wTW-' = lT,which proves proposition c. Theorem 3 Let I denote the identity matrix, w a Kvector of relative population sizes, and 1 a K-vector each element of which is equal to unity. Then the matrix H = I - w l T has the following properties: a. H2 = H ; i.e., H is idempotent. b. W H W - ' = H. c. HW-' = (HWP1jT; i.e., H W - l is symmetric. d. H W - l H = W-'H. e. tr(Hj = K - 1, where tr( ) denotes the trace, or summed diagonal elements of its argument. a. H ' H. = Proof I - 2wlT + wlTwlT = I - w l T = words, the second and higher e1,genvalues of M1 are also eigenvalues of L1, and the associated right eigenvectors are the same for both matrices. d. For j > i, vfLi = uJ%i = A j u j In other words, proposition c also holds for the left eigenvectors. e. L' has the same eigenvalues and eigenvectors a s MLexcept that the eigenvalue associated with the dyad w l T is unity in M', but zero in L'. Proof a. The definition of Li implies that Liw = M'w - wlTw. But lTw = 1,since the entries of w sum to unity, and M'w = w, since w is a right eigenvector of M with eigenvalue unity. Thus, L'w = w - w = 0, as claimed. b. The proof of proposition b parallels that of pro osition a, and follows from the fact that 1 is a left eigenvalue of M with eigenvalue unity. c. Since uj is a right eigenvector of M, it follows that Muj = Ajuj. If, in addition, we suppose that uj is not proportional to w, then the orthogonality of mismatched left and right eigenvectors ensures that lTuj = 0. Therefore, Lyu = M'uj - wl'uj = Xjuj, as claimed. d. Proposition d is proved in the same manner, using the fact that, if uj is not proportional to lT,then uyw = 0. e. This is merely a restatement of propositions (a) through (dj. F b. The result is obtained by substituting WHT = W - wwT into the left side of proposition b. c. Premultiply proposition b by W - l and then transpose. d. Proposition c implies that H W - l H = W-lH", from which d follows because of a. e. The trace of H is equal to t4I) - t d w l q , where the first of these traces equals K , and the second unity. Theorem 5 VXziVT, where is a diagonal matrix containing the eigenvalues of I,. (Li) n(Y-'Lk = Proof Theorem 4 allows Li to be written as W"ShLVT.Thus, cv;iisTw'/")w-1~'/'s;iiv~. ( ~ i ) ~ - = l ~ i Theorem 4 The theorem follows from the observation Let L' denote the matrix M'H, for i = 0,1,..., and X, the ith eigenvalue of M, indexed so that SW"W-'W'/"S = 1, since S is a n orthat A 1 > X, 2 ... 2 . ,X Denote by vyand thogonal matrix.

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