gene flow in natural populations es. izt3- 31s-

G. 1984. Carbon, water and nutrient rela-tions of two mistletoes and their hosts: ahypothesis. Plant Cell Environ. 7:293- 14999

133 Seigler, D. 1977. Plant systematics andalkaloids. In Alkaloids: Chemistry and 150Physiology, ed. R. H. F. Manske, 16:1-82

134. Shaner, D. L.. Boyer, J. S. 1976. Nitrate 151reductase activity in maize (Zea mays L.)

leaves: I. Regulation by nitrate flux.Plant Physiol. 58:499-504

135. Shaver, G. 1981. Mineral nutrition andleaf longevity in an evergreen shrub.Ledutn pa/ustre ssp. decumhens. Oecolo-gia 49:362-65 152.

136. Shaver, G., Melillo, J. 1984. Nutrientbudgets of marsh plants: efficiency con-cepts and relation to availability. Ecology 153.65:1491-510

137. Silvertown, J. W. 1982. Introduction toPlant Population Biology. New York: 154.Longman. 209 pp.

138. Small, E. 1972. Photosynthetic rates inrelation to nitrogen recycling as anadaptation to nutrient deficiency in peatbog plants. Can. J. Bot. 50:2227-33

139. Staaf, H. 1982. Plant nutrient changes in 155.beech leaves during senescence as in-fluenced by site characteristics. Aces

Oecologia/Oecologia Plant. 3:161-70140. Stachurski, A., Zimka, J. 1975. Methods 156.

of studying forest ecosystems: leaf area,leaf production and withdrawal of nutri-ents from leaves of trees. E:kol. Polska23:637-48 157.

141. Stark, N.. Steele. R. 1972. Nutrient con-tent of forest shrubs following burning.Ant. J. Bot. 64:1218-24

142. Stearns, S. 1976. Life-history tactics: areview of ideas. Q. Rev. Biol. 51:173-217 158.

143. Stewart, C., Hanson, A. 1980. Prolineaccumulation as a metabolic response towater stress. See Ref. 45. pp. 173-89 159.

144. Summerfield, R., Ricicy, J. 1973. Sub-strate freezing and thawing as a factor inthe mineral nutrient status of mire ecosys-tems. Plant Soil 38:557-66

145. Tanner. C. B., Sinclair, T. R. 1983. Effi-cient water use in crop production: re- 160.search or re-search? See Ref. 56. pp.1-27 161.

146. Taylor. B. 1967. Storage and mobiliza-tion of nitrogen in fruit trees: a review. J.Aust. Inst. Agri. Sci. 33:23-29

147. Thornlcy, J. H. 1972. A balanced quan-titative model for root-shoot ratios in 162.vegetative plants. Ann, Bat. 36:431-41

148. Tryon, P., Chapin, F. Ill. 1983. Tem-perature controls over root growth and

root biomass in taiga forest trees. Can. J.For. Res. 13:827-33

• Tukey. H. Jr. 1970. The leaching of sub-stances from plants. Ann. Rev. Plant.Physiol. 21:305-24

. Turner, J. 1977. Effect of nitrogenavailability on nitrogen cycling in aDouglas-fir stand. For. Sci. 23:307-16

• Van Cleve. K., Viereck, L. 1981. Forestsuccession in relation to nutrient cyclingin the boreal forests of Alaska. In ForestSuccession: Concepts and Applications,ed. D. C. West, H. Shugart, D. B. Bot-kin, pp. 184-211. Berlin: Springer-VerlagVan Den Driessche, R. 1974. Predictionof mineral nutrient status of trees by foliaranalysis. Bat. Rev. 40:347-94Vitousek, P. 1982. Nutrient cycling andnutrient use efficiency. Ant. Nat. 119:553-72Vogt. K., Edmonds, R.. Grier, C. 1981.Seasonal changes in biomass and verticaldistribution of mycorrhizal and fibrous-textured conifer fine roots in 23- and 180-year-old subalpine Abies amabilis stands.Can. J. For. Res. 11:223-29Wallsgrove, R. M., Keys, A. J., Lea, P.J.. Miflin, B. J. 1983. Photosynthesis,photorespiration. and nitrogen metabo-lism. Plant Cell Environ. 6:301-9Watson, M. A., Casper. B. B. 1984.Constraints on the expression of plantphenotypic plasticity. Ann. Rev. Ecol.Sm. 15:233-58West, N. E. 1981. Nutrient cycling indesert ecosystems. In Arid Land Ecosys-tems.: Structure, Functioning and Man-agement, ed. D. W. Goodall. R. A. Per-ry, 2:301-24. Cambridge: CambridgeUniv. PressWhite. L. 1973. Carbohydrate reservesof grasses: a review. J. Range Manage.26:13-18White. R. 1972. Studies on mineral ionabsorption by plants. I. The absorptionand utilization of phosphate by Stylo,san-thes Phaseohts atroputpureusand Desmodium intortum. Plant Soil36:427-47Wyn-Jones, R. G.. Gorham, J. 1982.Osmoregulation. See Ref. 77, pp. 35-58Youngner, V. 1972. Physiology of de-foliation and regrowth. In The Biologyand Utili:ation of Grasses, ed. V. B.Younger. C. M. McKell, pp. 292-303.New York: AcademicZiegler. H. 1975. Nature of transportsubstances. In Transport in Plants.Phloem transport. Encyclopedia of PlantPhysiology (NS). ed. M. H. Zimmer-man. J. A. Milburn, 1:59-136. Berlin:Springer-Verlag

39:' BLOOM, CHAPIN & MOONEY

Ann. Rev. Ecol. Syst. 1985. 16:393-430Copyright © 1985 by Annual Reviews Inc. All rights reserved

GENE FLOW IN NATURAL

POPULATIONSgefAA- es. Izt3- 31s- ckAL

Montgomery Slatkin* . /1 2-2.-

Department of Zoology, University of Washington, Seattle, Washington 98195

INTRODUCTION

Gene flow is a collective term that includes all mechanisms resulting in the

movement of genes from one population to another. Gene flow generally occurswithin a species, but examples of interspecific gene flow are known (36).Migration is sometimes used synonymously with gene flow, which is correctwhen migration between established populations is the mechanism of gene

flow. Gene flow can also be due to the movement of gametes, the extinction andrecolonization of entire populations, or the movement of extranuclear segmentsof DNA, such as mitochondria, plasmids, and viruses. Whatever the mech-anism, gene flow within a species largely determines the extent to whichgenetic changes in local populations are independent. In this article, I (a)review briefly the theoretical implications of different amounts of gene flow,(b) review methods for estimating levels of gene flow in nonhuman pop-ulations, and (c) discuss the evolutionary implications of some currently avail-able estimates. I do not discuss gene flow in humans, a large topic that has beenreviewed recently by Wijsman & Cavalli-Sforza (128).

Opinions about the evolutionary importance of gene flow have changed overthe past 30 years. Mayr (79, 80) advocated the concept of a "biologicalspecies," a group of organisms that are actually or potentially interbreeding. Heargued that the potential for interbreeding is often realized and that gene flow isimportant in maintaining the genetic and phenotypic homogeneity of a species.

*Current address is Department of Zoology, University of California, Berkeley, California94720

393

0066-4162/85/1120-0393$02.00

394 SLATKIN

GENE FLOW .395

although he did not claim that it is the only such mechanism. This became theaccepted view, but it was challenged by Ehrlich & Raven (27), Endler (29), andothers on two grounds. First, observations of dispersal showed that, for mostspecies, the actual movement of individuals was over distances much smallerthan individuals were capable of moving; second, sufficiently strong naturalselection could overcome the homogenizing effects of gene flow and producelocal differentiation. Ehrlich & Raven's conclusions were reinforced by otherstudies and theirs became the new prevailing opinion, although some of theevidence supporting their conclusion has been questioned by Jackson & Pounds(48) and others. Even Mayr (81) reduced his emphasis on gene flow, althoughmore recently Stanley (120) has revived Mayr's earlier conclusion and arguedthat gene flow may be one cause of morphological stasis.

I think it is time for another reappraisal of gene flow. The variety of methodsnow available for estimating levels of gene flow in natural populations showsthat levels vary widely among species. The patterns in gene frequencies fromnatural populations strongly suggest that species differ greatly in the extent ofgene flow they experience. In some species, the patterns are consistent withlevels of gene flow inferred from direct observations of dispersal. For suchspecies, we have confidence that we do understand the mechanisms and level ofgene flow. For other species, different methods lead to different conclusions,with gene frequency data implying a substantially higher level of gene flow thando direct observations of dispersal. For such species, the lack of concordanceindicates the highly unpredictable character of gene flow. Gene flow under"normal" conditions may indeed be restricted, but, under other conditions,particularly those that cause large-scale demographic changes, gene flow overlong distances will occur. For some species, gene flow resulting from theextinction and recolonization of local populations may be the principal mech-

anism of gene flow.. The differences in levels of gene flow in different species mean that geneflow does not have a single evolutionary role and probably does not account formorphological uniformity of most species. Instead, gene frequency data anddirect observations of dispersal under normal conditions together can providesome insight into the demographic properties of different species.

GENE FLOW AS A HOMOGENIZING FORCE

How much gene flow is necessary to prevent genetic evolution from beingindependent in different populations depends on what other mechanisms ofgenetic change are at work. Except for that resulting from the movement ofextranuclear DNA, gene flow usually affects all nuclear loci in the same way(12), but each genetic locus and each phenotypic character can be governed bydifferent selective regimes and different mutational processes. As a result,

some loci may show little geographic differentiation while others show extremedifferentiation.

Models of Population Structure

Mathematical models make clear predictions about the interaction of gene flowwith other forces of genetic change but only under specific assumptions aboutthe geographic and demographic structure of populations. Two large classes ofpopulation structures are assumed: those with a population comprising discretesubpopulations and those with a continuously distributed population. Althoughthese two types of models appear different, their predictions are consistent ifthey are properly formulated.

Models of a continuously distributed population have an inherent geographicstructure. Every location is identified by a coordinate in a one-, two-, orthree-dimensional system. In these models, information about gene flow iscontained in the distribution of distances between locations of the birth of anindividual and of its parent or parents, i.e. the distribution of dispersal dis-tances. In models of finite, continuously distributed populations, the popula-tion density—the number of individuals per unit area—must also be specified.In most models, uniform population densities are assumed and the samedistribution of dispersal distances is assumed at each location. In such models,the variance of the dispersal distances, o- 2 , is of particular interest, a pointdiscussed later.

Models with discrete subpopulations do not have an inherent geographicstructure in the sense that the physical locations of subpopulations may heunknown or unimportant. What is important is the extent of gene flow betweensubpopulations. It is common to assume discrete nonoverlapping generations,in which case the level of gene flow between a particular pair of subpopulationsis defined to be the average fraction of individuals in each subpopulation in eachgeneration that is derived from a particular other subpopulation. For a modelwith n subpopulations, this information is summarized in an n x n matrix,called the "migration matrix" (8) and often denoted by Mu. This definition is ofthe "backward" matrix because it describes events in the previous generation.The "forward" matrix, AP ii , predicts the fraction of the offspring in population iin one generation that will breed in population j in the next generation. Unlessthe movements among subpopulations are exactly balanced, with equal num-bers going in each direction, the forward and backward matrixes are different.For mathematical convenience, though, that symmetry is usually assumed inmodels.

While an arbitrary migration matrix can be specified, very few usefulconclusions about gene flow can be reached without assuming a particularpattern of gene flow. For mathematical convenience, several regular patternsare assumed. One is the island model, first used by Wright (l30). Originally,

396 SLATKIN GENE FLOW 397

the island model represented a subpopulation in an infinite collection of equiv-alent subpopulations. The subpopulation of interest receives immigrant in-dividuals at a rate m chosen at random from the other subpopulations. Theassumption of an infinite number of subpopulations ensures that the averageproperties of the collection do not change, making it necessary to model onlyone subpopulation because each one is representative of the entire collection. Iwill refer to this model as the infinite island model. In more recent usage, theisland model represents a finite number, n, of subpopulations, each of which isequally likely to send individuals to and from every other (60). To distinguishthis model from the infinite island model, I will call it the n-island model. Thereis no geography in either island model because every subpopulation is equallyaccessible to every other. The island model, then, represents the extreme inlong-distance gene flow.

Another simple population structure is one in which subpopulations arelocated in a regular one-, two-, or three-dimensional lattice, with the samefraction of individuals moving between every pair of adjacent Populations. Thisis the stepping-stone model introduced by Kimura (55) and named for the use ofstepping-stones in Japanese gardens (M. Kimura, personal communication). Inone dimension, gene flow is between a subpopulation and two immediateneighbors; in two dimensions, gene flow is between a subpopulation and itsfour or six immediate neighbors depending on whether a rectangular or hex-agonal lattice is assumed. The stepping-stone model represents the extreme inshort-distance gene flow. In using a stepping-stone model of population struc-ture, it is common to assume either an infinite number of subpopulations or toassume that subpopulations are arranged in a circle (in one dimension) or a torus(in two dimensions). Either of those assumptions eliminates the possibility of"edge effects" caused because some subpopulations are closer to boundariesthan others.

Gene Flow and Genetic Drift

ISLAND MODELS Genetic drift caused by the random sampling of genes at thetime of zygote formation results in local genetic differentiation unless it isopposed by gene flow. Like gene flow, genetic drift affects all nuclear loci to_the same degree. Using the infinite island model, Wright (130) showed that if afraction, tn, of a population is replaced by immigrants, there will be nosignificant genetic differentiation if Nm > 1, where N is the effective size of thepopulation. More precisely, Wright showed that if the frequency of a particularneutral allele in the immigrants is p, the average frequency in the sub-populations is also p, and the variance in frequency among subpopulations isp(1—p)/(1 +4Nm) when m is small. As in approaches 0, the varianceapproaches its maximum possible value, p(1 —p), indicating that a fraction, p,

of the populations is nearly fixed for the allele of interest and a fraction. ( I — p).is nearly fixed for other alleles. Wright's result is the basis for his rule that oneimmigrant individual every other generation is sufficient to prevent dil-

1 ferentiation due to genetic drift. As the formula for the variance in frequency! indicates, the transition from a small to a large degree of differentiation does not

occur abruptly at 2Nm I , but Wright's rule does provide a convenientsummary of the interaction between gene flow and genetic drift in the infinite

1 island model. Wright's conclusion about the level of gene flow needed toprevent genetic differentiation has been confirmed in other analyses of islandmodels and stepping-stone models. The literature in this area has been reviewedby Felsenstein (33) and Nagylaki (88, 90).

In models such as the n-island model that represent a finite number ofsubpopulations, all genetic variability can be lost. In the absence of mutation.the only equilibrium state of a locus is the fixation of one allele throughout thepopulation. Two ways of examining genetic differentiation in such models havebeen used. One is to assume no mutation and find the spatial patterns in allelefrequencies before fixation of one allele; the other is to assume mutations dooccur and find the extent of differentiation under both mutation and gene flow.Both kinds of models have interesting mathematical properties, but the resultsare not always consistent. Models including mutation are more directly in-terpretable because all genetic loci are thought to be mutable.

Models of gene flow, genetic drift, and mutation are usually formulated interms of probabilities of identity for pairs of alleles. This approach wasdeveloped independently by Malécot (70) and Cotterman (15), although its usetraces to Malecot's influential, bofok (70). In the "infinite alleles" model ofmutation (56), in which each allele has a probability, R, per generation ofmutating to an allele that is new to the population, alleles are identical only ifthey are descended from the same mutant, so the probability of identity of twoalleles is the probability they are identical by descent. In a model with a finitenumber of alleles, in which every allele has the same probability of mutating toevery other allele, the probability of identity of two alleles is the probabilitythey are identical in state, because alleles can be identical in state without beingidentical by descent. In practice, the distinction between identity in state andidentity by descent is not usually made or needed. I will follow the commonpractice and use the same symbol, f, for both.

In the n-island model, all subpopulations and pairs of subpopulations areequivalent, so it is sufficient to use the probabilities of identity of two allelesrandomly chosen from the same subpopulation, f0 , and from two differentsubpopulations, Latter (60) derived the solution for the equilibrium andtime-dependent properties of the n-island model, generalizing the results ofMaruyama (74) and Maynard Smith (78). Recently, Crow & Aoki (17) and

IJ

6

!T


Takahata & Nei (122) presented essentially the same results as Latter's but in adifferent notation. At equilibrium, one way to express the extent of differentia-tion in the n-island model is: vrc- °°`-'

1-3

(fo — j)1(1 — 11(4Nmet + I), 1.

if µ«m, where f [fo + (n— 1 )f ]in is the average probability of identity oftwo alleles drawn at random from the population and a = [n/(n— 1)1 2 . Althoughthis model includes mutation, Equation 1 reduces to Wright's formula for theinfinite island model in the limit as n becomes large.

A different way to describe the equilibrium results from the n-island modelwith mutation is the ratio f l /fo, which measures the relative probabilities ofidentity within and between subpopulations (60):

Tr4-4,-Lvfi /fo = 141 + (It — 1) vim]. 2. a 0

Note that Equation 1 depends on Nm and not on p. whereas Equation 2 dependson Wm but not N.

Equation I provides information about the relative probabilities of drawingalleles that are not identical from subpopulations that are different and fromones that are the same; Equation 2 provides information about the relativeprobabilities of identity. As Latter (61) has pointed out, these two ways ofmeasuring population subdivision contain different and not necessarily relatedkinds of information. These two measures correspond to different kinds ofgenetic distances: Equation 2 is a quantity that is estimated by genetic auto-correlation analysis and by the kind of genetic distance proposed by Nei (93);Equation 1 is a quantity estimated by Wright's FST and by the kind of geneticdistance proposed by Caval!i-Sforza & Edwards (13). Some discussion hasoccurred about which of these two kinds of measures of population differentia-tion is more useful (e.g. 33 : 271). These formulae suggest that within a singlespecies the choice depends on whether the concern is with the relative im-portance of gene flow and genetic drift or with the relative importance of geneflow and mutation.

There is a third combination off and fo that has still other properties. Crow &Maruyama (18) showed that at equilibriumf = (1 — fo)/(4Nµ), for an arbitrarymigration matrix. This result is not independent of the other two and can bederived from Equations 1 and 2 for the n-island model.

STEPPING-STONE MODELS The results from stepping-stone models are sim-ilar in some ways to those from island models, although the similarities are notalways emphasized. Malecot (70) and Kimura & Weiss (58) analyzed infinitestepping-stone models, and Malecot (72) and Maruyama (74, 75) analyzedfinite one- and two-dimensional stepping-stone models. The theory in this areais conveniently summarized by Malecot (72). In an infinite one-dimensional

stepping-stone model with mutations occurring at a rate p., the results are in

terms of f----the probabilities of identity of two alleles drawn from sub-populations i steps apart. As in the case of the n-island model, there are twomeasures of differentiation. Kimura & Weiss (58) showed:

fi lfo = I — V 2p./m, 3.

when p,<<m, where m/2 is the rate of gene flow between adjacent sub-populations. This result is similar to Equation 2 in its dependence on the ratiop./m. On the other hand,

fo) = 1/(4Nm), 4.

when m>>41., which is similar in form to Equation I.For subpopulations i steps apart, f is proportional to expl —iV(2p/ni)j.

which implies that subpopulations separated by more than 1,. = (m12p..) stepseffectively share few alleles. That this is a "characteristic length" associatedwith gene flow and drift in this model is supported by the results of Malecot (72)which show that finite stepping-stone models with many more than /, sub-populations have properties similar to those of the infinite stepping-stonemodel, but models with fewer than 1, subpopulations are essentially panmictic.It is worth noting that /, is the average distance that a new mutation is likely todiffuse in the population before it mutates. For subpopulations farther apart, thedistinction between the two measures of genetic differentiation is less signifi-cant. Both fi/f0 and (fo — fi)/(1 — fo) decrease in proportion to exp(—i//, ).

The results from a one-dimensional continuum model are similar to thosefrom a one-dimensional stepping-stone model. If o- 2 is the variance in dispersaldistances,f(x), the probability of identity of two alleles chosen from a distance x

I apart, is proportional to exp[—(x/cr)V(201. In this model and the correspond-ing two-dimensional model, only the variance in dispersal distances, a2 , affectsthe rate of decrease in the probability of identity with distance in a population at-equilibrium, assuming the average dispersal distance is zero. Other propertiesof the distribution of dispersal distances, such as the degree of kurtosis, do notaffect this result.

In a model of a continuously distributed population, there is nothing compa-rable to an adjacent subpopulation, but we can find the degree of differentiationat x = a, i.e, at roughly one dispersal distance. From Malecot [72, Equations(55, 56)], for a habitat of infinite size,

f(a)If(0) 1 — \/(2p.) 5.

when p.<<1, which is equivalent to Equation 3. As expected, the ratiocorresponding to Equation 4 has different properties:

[f(0) — f(cr)]/[1 — f(0)] = 1/(4Dcr), 6.


where D is the number of individuals per unit length.Wright's (131) analysis of a continuously distributed population was based

on the assumption that the neighborhood size, defined to be the number ofindividuals within one dispersal distance, determined the extent of geneticdifferentiation. His results were derived under the assumption of no mutation,so they are not comparable to those reviewed here. Nevertheless, neighborhoodsize is commonly computed and commonly assumed to measure the potentialfor local genetic differentiation. The results discussed above show that is not thecase, as Malecot has pointed out (e.g. 71:76) and as Rohif & Schnell (105)confirmed in a simulation study. The increase in differentiation with distance atequilibrium depends on o-/Vµ„ not on the neighborhood size, Do-, which entersonly in Equation 6. Furthermore, the continuum model is reasonable only if Do->> I. Otherwise many individuals would not be dispersing far enough to reachtheir nearest neighbors, creating what might be thought an awkward breedingsystem. Equation 6 implies, then, that when the continuum model can be usedat all, there is essentially no differentiation between locations separated by adistance o-.

In two-dimensional stepping-stone models, the results are similar to thosefrom one-dimensional models but are more difficult to obtain mathematically.For the infinite stepping-stone model, I will use the results of Kimura & Weiss(58) but interpret them in terms of probabilities of identities descent. Forsubpopulations separated by i i steps in one direction and i 2 steps in the other, leti = \/(i, 2 + i2 2 ) be the net distance between them. If the rate of gene flowbetween adjacent subpopulations is m/4, then, if p,<<m<< I ,

fi N = 1 - rr in[4V(viltn)] I 2; 7.

and

- fi )1(1 — = 1/14Nurril1n[41/(2p./m)]}. 8.

Equation 7 is similar to Equations 2 and 3 in that it depends only on Rim.Equation 8 differs from Equations 1 and 4, however, in that 7 depends on bothNm and RItn, but the result is qualitatively similar because the dependence onRItn is weak. For large values of f, is proportional to exp[—/1/(8R/m)]/Vi,which implies a more rapid decrease with distance than in a one-dimensionalmodel. The characteristic length associated with gene flow and drift is pro-portional to V(m/p,), as in the one-dimensional model. If both the width andlength of a finite stepping-stone model are small with respect to this length, thenthe population is effectively panmictic. If the width of a two-dimensionalstepping-stone model is much smaller than 4., the model is effectively one-dimensional.

Two-dimensional models of a continuously distributed population lead toserious mathematical difficulties. The decrease in identity for locations sepa-

rated by large distances is approximately the same as in the two-dimensionalstepping-stone model. For short distances, however, mathematical singulari-ties arise (88). It is not clear at present whether these mathematical problems aredue to the inadequacy of the mathematical methods used for finding probabili-ties of identity or whether the problems are a manifestation of the demographicproblems discussed by Felsenstein (31). In any case, there are no formulaecomparable to Equations 5 and 6 for the two-dimensional continuum model.

Natural populations do not exist in perfectly regular one- or two-dimensionalarrays. These conditions are assumed only for mathematical convenience.Populations probably do differ in the extent to which their ranges are morenearly approximated by one- or two-dimensional models. A range that is longand narrow is effectively one-dimensional even though it is not literally so. Theresults described above indicate that at the same geographic distance, moregenetic differentiation is expected in two dimensions than in one. This seems tocontradict the conclusion of Kimura & Maruyama (57) and others that moredifferentiation is expected in one dimension. Their conclusions, though, arebased on models in which there is no mutation. Before mutation becomesimportant, models assuming no mutation describe expected patterns of dif-ferentiation, but the theories discussed above show how that is not true forlonger times.

EXTINCTION AND RECOLONIZATION Most models of gene flow assume thepopulation structure is fixed and gene flow is due to the movement of gametesor individuals between locations. Gene flow can also result from the estahl ish-ment of new subpopulations. For species in which the extinction andrecolonization of subpopulations are frequent. this mechanism could easily bean important cause of gene flow. There are only a few models of the geneticconsequences of extinction and recolonization (76. 107, I 10). The importanceof extinction and recolonization is more commonly discussed by ecologists(69).

I analyzed an n-island model in which each local population had a probabilitye0 of going extinct each generation (110). When an extinction occurs, thevacant site is recolonized in the next generation by k individuals chosen fromother populations. My results can be rewritten to show that if eo is small andeo, Equations 1 and 2 describe the equilibrium results with m replaced by m +(eo/2). Therefore, even if m = 0, there will be little differentiation of sub-populations if 6,0 is large enough.

Gene Flow and Selection

ISLAND MODEL Haldane (41) first considered the balance achieved betweengene flow and selection in a single population. He showed that if an allele is

402' SLATKIN GENE FLOW 403

favored in a population by selection of strength s, then immigration of otheralleles at a rate m would not prevent a high frequency of the favored allele if,roughly, m < s. Like Wright's (130) rule (Nm>1), that is not a sharp thresholdbut a useful characterization of the results. Nagylaki (86) extended Haldane'sresults and emphasized that, under a balance between gene flow and selection,it is difficult to maintain an allele in low frequency. There is only a very narrowrange of parameter values in the model for whiefi the equilibrium frequency ofthe favored allele is low. For most parameter values, the favored allele is eitherswamped by immigration or has a high frequency.

strong ((3 of order one in magnitude), genetic drift makes little difference in the

shape of the cline expected in the absence of genetic drift, and selection changesonly slightly the genetic variance and correlation with distance expected in theabsence of selection. For the purposes of many evolutionary discussions, it isadequate to use the theories of gene flow and genetic drift and of gene flow andselection separately to describe the potential for the genetic differentiation oflocal populations. That is not always true, though. Barton (4) has shown that acline maintained by selection at one locus can act as a partial barrier to geneflow at neutral loci that are closely linked to the selected locus.

CLINES Models of gene flow and selection in a continuously distributedpopulation have been directed largely to understanding clines, a term in-troduced by Huxley (47) to describe a gradual change in the average of aphenotypic character with spatial location. The term has been extended togradual changes in allele frequency. A cline may appear abrupt when viewed onthe scale of the entire species, but the term implies there is some zone oftransition. Haldane (42) and Fisher (37) showed that permanent clines canresult from the balance between gene flow and selection.

In my analyses of clines I showed there is a minimum length scale—thecharacteristic length—in a cline due to gene flow and selection (109). Thecharacteristic length for gene flow and selection, lc , is o-/Vs, where o is thestandard deviation of dispersal distances and s is the average strength ofselection affecting alleles at the locus of interest. The characteristic length is ameasure of the minimum length of the scale of geographic change and isroughly the width of a cline if there is an abrupt change in selection intensities—a "step cline" (29). If the change in selection intensities is more gradual, theresulting cline is wider than the characteristic length (77). if there is a region inwhich one allele is favored surrounded by regions in which other alleles arefavored (an environmental "pocket"), the favored allele can be maintained if thepocket is larger than the characteristic length (44, 86, 109). Nagylaki (86)showed that, as in the island model, an allele favored in an environmentalpocket is likely either to be present in high frequency or absent. In general, it isvery difficult to maintain an allele at low frequency in one region through abalance between gene flow and selection.

The analysis of gene flow and selection in finite populations is much moredifficult mathematically than the case of selection or drift alone. Maruyama andI (116) and Felsenstein (32) tried different types of approximations, but thedefinitive analysis was done by Nagylaki (89, 91). He showed that the relativeimportance of genetic drift and selection in a step cline is determined by the \single dimensionless parameter, 13 = 2 DoiVs. Nagylaki showed that selection \is strong compared to drift when (3 << 1 and is weak compared to drift when 13>> 1. Nagylaki & Lucier (91) showed that even when selection is not very

INDIRECT METHODS FOR ESTIMATING GENE FLOW

An indirect method is one that uses observed spatial distributions of alleles.chromosomal segments, or phenotypic traits to draw inferences about the levelor pattern of gene flow in a population. In any indirect method, there are twosteps—computing various statistics from data, and relating those statistics toparameters of a model of gene flow and of other mechanisms of geneticevolution. Lewontin (68) argues that population genetics theory is ill suited forworking from data to the estimation of parameter values. As he illustrates withthree examples—the estimation of the average degree of dominance of aquantitative character, the estimation of fitness differences from viabilities, andthe estimation of effective population sizes from the frequencies of allelism oflethals—estimates of parameter values are too sensitive to forces of geneticevolution not accounted for in the particular estimation scheme for there to be r

, any confidence in the estimates. Lewontin argues that population genetics ;Vi -' I-theory is more correctly used to assess the plausibility of different mechanisms

141s explanations for observed patterns.Indirect methods of estimating the levels and patterns of gene flow attempt to

do what Lewontin claims is not possible. If Lewontin is correct, then all themethods described below will fail. I am less pessimistic and think that indirectmethods do hold promise for estimating levels of gene flow. There are obviouslimitations and the necessary theory is far from complete, but I am optimisticbecause gene flow and genetic drift affect all nuclear loci in the same way, asCavalli-Sforza (12) pointed out. Estimates based on data from one or two locishould be suspect, but if estimates are based on data from numerous loci and

(there is consistency in the estimates using different methods, it is reasonable tohave some confidence in the conclusions.

Allelism of LethalsThe first estimates of Nm were based on the frequency of allelic lethals. Theidea is that a recessive or partly recessive lethal is so strongly selected againstthat it will persist for only a short time. A pair of allelic lethals can reasonably be


assumed to have descended from the same mutation in the recent past. Simplemodels predict how the effective population size, heterozygote fitness, andimmigration rate will affect the probability of allelism of lethals from the sameand different populations.

Dobzhansky & Wright (22) found the frequency of allelism of lethal andsemilethal third chromosomes in isolated populations of Drosophilapsettdoobscura and used their observations to estimate several populationparameters. They estimated the value of Nm to be 54, where N is the effectivesize of a local population and in is the fraction of immigrants in each generation.In discussing possible biases of their method, they concluded that this estimatewas probably too large but that the true value was greater than 5. Wright et al(136) carried out a similar study of the same species in an area where it wasmore continuously distributed. They compared the frequency of allelism atdifferent collecting stations within each of three localities and between locali-ties. Stations within a locality were 0.2 to 3.5 km apart and localities were aminimum of 22 km apart. Wright et al (136) found that the frequency of allelismbetween different stations in the same locality was indistinguishable from thefrequency within a station and significantly different from the frequency ofallelism between localities. They did not produce a single estimate of Nm butconcluded that stations within a locality were not isolated from one another.

The allelism of lethals has been used in numerous other studies, particularlyin the 1950s and 1960s, but usually to estimate either the local population sizeor the relative fitness of individuals heterozygous for lethals, rather than thelevel of gene flow. This literature is reviewed by Crow & Temin (19) andWallace (126). Lewontin (68) discusses the difficulties with these methods.

Wallace (126) took a different approach to using allelic lethals and measuredthe decrease in frequency of allelism with geographic distance. Using thedistribution of dispersal distances found by Bateman (7) and Wallace (125), heassumed that the logarithm of the frequency of allelic lethals would decreaselinearly with the square root of distance. Wallace (126) fitted data on frequen-cies of allelism at distances of 0, 30, 60, and 90 m in D. melanogaster inColombia to show that an approximately 50% decrease in the frequency ofallelism would be expected at a distance of 150 m. Yokoyama (137) developedan analytic approximation and carried out some computer simulations to testWallace's conjecture about the dependence of allelism frequency on distance.Yokoyama found in a stepping-stone model that, for large distances, allelismfrequency decreases approximately as exp(—cd) in one dimension and as[exp(—cd)]/Vd in two dimensions, where d is distance and c is a constant thatdepends on population size, fitness of heterozygotes, and the migration rate.The dependence on distance is then different from what Wallace had suggested.The mathematical form of these results is the same as that found by Malecot(71) and Kimura & Weiss (58) for neutral alleles in one- and two-dimensional

stepping-stone models. Yokoyama showed that, at a given distance, the ex-pected frequency of allelism is higher in one dimension than in two. Yokoyamadid not develop a method for estimating from data the rate of gene flow, but hecompared his model's predictions for different parameter values with data ofWallace (126) and similar data' of Paik & Sung (99). Yokoyama plotted theresults for a one-dimensional stepping-stone model with subpopulationsassumed to be 30 m apart. He found good agreement with the data when N =

100, m 0.5, and the lethal is completely recessive. He saw a much worse fitwhen N was larger or m smaller. Because the decrease in allelism frequencywith distance is more rapid in two dimensions than in one, this has to heregarded as a minimum estimate of in. Larger values of in cannot be takenliterally because that would imply that more than half of each population ismade up of immigrants each generation. Instead, the conclusion is that signifi-cant gene flow must be occurring over distances much greater than 30 m: this isconsistent with the results of direct measurements of dispersal rates in Dro-sophila, a point I discuss later.

FST and Related Quantities

The literature in population genetics abounds with definitions of inbreedingcoefficients, fixation indexes, and related quantities. The confusion surround-ing terminology in this area is one of the main impediments to the effective useof these quantities. Quantities related to inbreeding coefficients can be used toestimate the average level of gene flow under the assumption that alleles areneutral, although statistical problems in this area are far from resolved.

Wright (129) defined the correlation coefficient of an allele. F, to be thecorrelation in the presence or absence of that allele in uniting gametes (see133: 173-74). If the genotype of every individual in a very large population isknown, F = (P — p2)/[p(1 — p)], where p is the frequency of the allele ofinterest and P is the frequency of individuals homozygous for that allele. Ifthere are more than two alleles at a locus each may have a different value of F.although those values are not independent. Modifications of this formula areneeded for small populations or for samples from populations, a point discussedby Robertson & Hill (104).

As Wright emphasized (e.g. 134 : 60), the correlation coefficient, which iscomputed from allele frequencies, is a statistic and differs from both theinbreeding coefficient and the probability of identity. Those quantities are notstatistics and not directly measureable. They are derived from populationgenetic models, and their values can be estimated from F if suitable assump-tions are made. The inbreeding coefficient, as initially defined by Wright (129),is the expected correlation between uniting gametes. Therefore, if no otherforces, such as mutation and selection, are affecting a locus, the expected valueof F, the correlation coefficient for each allele, is the inbreeding coefficient.

406 SLATKINGENE FLOW 407

Observed values of F can then be used to estimate inbreeding coefficients underthe assumption of neutrality. Unfortunately, Wright usually used F for both thecorrelation coefficient and the inbreeding coefficient. In his early papers heusedf for the inbreeding coefficient (e.g. 129), but at least since his 1931 paper(130), he has used F. In his early papers, he was concerned with the analysis ofpedigrees and called the inbreeding coefficient computed from pedigrees the"correlation;" thus he did not distinguish between the correlation coefficientand its expectation.

The probability of identity of pairs of alleles----either identity in state oridentity by descent—is also a quantity derived from population genetic modelsand is useful when considering neutral alleles. It differs slightly from theinbreeding coefficient because it is defined for a particular locus, whereas theinbreeding coefficient is defined for all loci in uniting gametes. If there are noother forces affecting a locus, the probability of identity of two alleles in unitinggametes is the same as the inbreeding coefficient. The probability of identitywill differ from the inbreeding coefficient if mutations are allowed in themodel. A mutation reduces the probability of identity but does not change theinbreeding coefficient.

For a population divided into subpopulations, Wright (132) partitioned F intocomponents. He defined FIT to be the correlation coefficient between unitinggametes relative to the whole populations, F Is to be the correlation coefficientof uniting gametes relative to alleles in a particular subpopulation, and FST to bethe correlation of two randomly chosen alleles in a subpopulation relative toalleles in the whole population. I will use these symbols for the correlations,although Wright used them for both the correlations and their expected values.Because of the confusion in notation Weir & Cockerham (127) advocate usingdifferent symbols completely. For neutral alleles, FIT measures the extent ofinbreeding in the entire population, F Is measures the inbreeding due to nonran-dom mating in each subpopulation, and FST measures the inbreeding due to thecorrelation among alleles caused by their occurrence in the same subpopula-tion. If there is random mating in each subpopulation, as is usually assumed inmodels of subdivided populations, F = 0, implying F/T = F sr.

Wright (132) showed that, for an infinite number of subpopulations and two

alleles at a locus, FST = 17,1[13(1 — p)], where p and V„ are the mean and varianceamong subpopulations of the frequency of either allele. This result requires noassumptions about the causes of differences among subpopulations. For neutralalleles in the infinite island model, FST = 1/(1 + 4Nrn), and in this case there isno need to distinguish between F sy and its expectation because p and V, arecertain to take their expected values. This result for FST in the infinite islandmodel suggests that observed values of FST can be used to estimate Nm, and thathas been attempted by Lewontin (67 : 213), Larson et al (59), and others. I think

that is justified, but some care is needed to understand exactly what assump-

tions are being made.There are two sources of variation in values of F sy computed both for

different alleles at a locus and for alleles at different loci. One is the variationdue to sampling individuals from a population and the other is due to differencesamong alleles and among loci in the values of F 57- in the population from whichsamples are taken. If it is assumed that the underlying values of Fs-, for allalleles at a locus are the same, with differences among observed values due onlyto sampling, then the problem is to combine information from different allelesto produce an estimate of FST with the best statistical properties. Robertson &Hill (104) and Weir & Cockerham (127) show that different weightings of Fs).

values for different alleles have different statistical properties. Weir & Cock-erham (127) use simulations to compare several methods and show that the onedeveloped by Reynolds et al (102) has the least bias and the smallest variance ofthe five methods they examined. This method accounts for finite sample sizes.finite numbers of subpopulations, and the variance in sample sizes amongsubpopulations.

With large sample sizes, large numbers of subpopulations, and no variance insample sizes, Reynolds et al's (102) weighting of values from different alleles isthe same as that used by Nei (94) in his definition of G sy , which is the same asthe weighting used by McPhee & Wright (83) in their analysis of pedigrees inshorthorn cattle. This is especially convenient because the expected value ofGsr can be related to the predictions of the n-island model:

G ST = VO — .n/(1 — .fl. 9.

From Equation 1, the right hand side is 1/(1 + 4Ntn a), where a = [111(1— 1)12.

Thus, Wright's formula for F sy derived for the infinite island model withoutmutation is approximately correct for the n-island model with mutation if n is

large and t.t, is small.Nei's (94) formula for computing FST (i.e. G sT) from allele frequencies does

not take account of sampling, so it is better to use the method proposed byReynolds et al (102) to estimate F sy and then estimate Nm from

(Nm),„ = (11FsT — 1)/4. 10.

The resulting estimate is the value Nm would have if the population structure

were an island model and if the actual value of FST were the same for every

allele. It is an underestimate if the true population is similar to a stepping-stonestructure, although Crow & Aoki-(17) show that a two-dimensional steppin g

-stone model produces results similar to the n-island model. If there are ex-tinctions and recolonizations occurring, my results for an n-island model (110)

408 SLATKIN GENE 11.0W 409

show that this procedure estimates N(m + e0/2), where e0 is the extinction rateper subpopulation per generation, instead of Nm.

This method of estimating Nm seems reasonable under the assumption ofneutrality. Equation 1 shows that the expected value of FsT is the same foralleles at loci with different mutation rates. This means that allele frequenciesfrom different loci can be combined without having to assume equal mutationrates. That contrasts to approaches using some genetic distances and is dis-cussed below. The accuracy of this method has not been tested, however, andthe testing will have to be done, probably by simulation, before the results arecompletely interpretable. While Weir & Cockerham's (127) analysis doesaccount for sampling, it does not account for differences in the history of allelescaused by the stochastic nature of gene flow, mutation, and drift. Because of

3those differences in history, the actual values of F sT in the population would notbe the same for different alleles even though their expectations are the same.

How much effect differences in history have on this method for estimatingNm is at present unknown. Takahata (121) has shown for the n-island modelthat the variance in the identity of genes sampled from the whole population islarge unless Nm>> 1. It is instructive and not particularly encouraging toreview the results for a similar case for which the answer is known. In a singlepanmictic population, the probability of identity by descent under mutation anddrift is f = 1/(1+4 Nti.) (71). The same logic as used above for estimating Nmsuggests that an estimator of 4Np. is (1/H — 1), where H Exi and the xi areallele frequencies in a sample. Ewens (30) has shown, however, that, under theassumptions of the neutral mutation model, this approach does not lead to eitheran efficient or an unbiased estimate of 4Npu, and that a better estimator iscomputed from k, the number of distinct alleles in the sample. There seems noreason to suppose that the problem of estimating Nm is any easier.

Rare AllelesI have developed a different approach to estimating the average level of geneflow by using the distribution of rare alleles, alleles that are not found in everysubpopulation. This method is based on the properties of the "conditionalaverage frequencies," the average frequencies of alleles found in some but notall locations sampled (I 1 1 , 112, 114). If there are d locations sampled, theconditional average frequencies, denoted by p(i), are the average frequencies ofall alleles found in exactly i of the d locations. Of particular interest is p(i), theaverage frequency of alleles found in only one location, which are called"private" alleles (92). Rare alleles are particularly sensitive to gene flowbecause it is unlikely that rare alleles will be carried by dispersing gametes orindividuals unless dispersal is frequent. The potential utility of rare alleles asindicators of gene flow was noted by Pashley & Bush (100).

I first examined the conditional average frequencies because of Ewens's (30)

sampling theory of neutral alleles. Ewens showed that the number of distinctalleles in a sample was sufficient to estimate 4Np. and that the distribution ofallele frequencies, given the number of distinct alleles, could provide a test ofneutrality. Working by analogy with Ewens's theory, I tried conditioning on thenumber of populations in which an allele is found, to find a test of neutrality thatcould be applied to subdivided populations. That proved impossible because Ifound p (i) to be nearly independent of the selection affecting the locus (111.112). They are also nearly independent of mutation rates (112) but dependstrongly on the average level of gene flow, as measured by Nm in a stepping-stone or island model. This suggested that the conditional average frequenciesmight be useful for estimating Nm because they could be averaged over lociwithout having to assume either equal mutation rates or complete neutrality.

I first used the p(i) to obtain a qualitative estimate of Nm by making visual'comparison of the results generated by a simulation program and valuescomputed from published data (112). I found some species, such as two

Drosophila species, to have values of p(i) similar to those from a model inwhich Nm> 1; other species including several salamander species had values ofp(i) similar to a model in which Nm<1. More recently, I found that the averagefrequency of private alleles, p(1), can provide a quantitative estimate of Nm

(114). Using this method, which includes an approximate way to correct fordifferences in sample size, I estimated values of Nm for several species andfound they differed by at least two orders of magnitude, with values rangingfrom 9.9 forDrosophila willistoni to 0.1 for Methadon dorscdis. Because of thelong time necessary for the conditional average frequencies to reach an equilib-rium for low levels of gene flow, this method cannot distinguish low levels ofcontinuing gene flow from no current gene flow.

Subsamples of PopulationsAny method that yields an estimate of Nm for all locations from which sampleswere taken can be applied to subsets of sample locations, yielding an estimate ofNm within that subsample. I used this approach in applying the rare allelemethod to find if there were any evidence for heterogeneity in levels of geneflow (114). I analyzed the data of Nevo & Yang (98) on the frog, Hyla arborea.

The estimate of Nm using 7 sampling locations is 0.50; when each 1 of 6 of 7sampling locations is omitted from the analysis, estimates range from 0.42 to0.54 (114, Table 8). When one of the locations is omitted, the estimate of Nm

increases to 6.2, suggesting that location, which is not at the periphery of thespecies range, is isolated or nearly so from the rest of the species.

Spatial AutocorrelationThe correlation of variables at different spatial locations is easily computed andis independent of any assumptions about population structure. Cliff & Ord (14)

410 ' SLATKIN GENE FLOW 41 I

summarized various methods of computing spatial autocorrelations and pop-ularized their use. Sokal and his coworkers have been primarily responsible forthe use of these methods in population biology (117-119).

The starting point in computing the spatial autocorrelation in allele frequen-cies is a list of frequencies at each spatial location. The locations are connectedby lines ("edges" in the terminology of graph theory), with the connectionsdetermined by proximity and possibly other criteria. Locations that are con-nected by edges are regarded as being one unit of geographic distance apart. Forlocations that are not connected by edges, an additional criterion for distance isneeded. One possible measure of distance is the minimum number of edgestraversed in going from one location to another; another possibility is thegeographic distance between locations. For each allele, the correlation coeffi-cient is computed for all pairs of locations that are a specified distance apart,using the product-moment correlation or a more general correlation coefficient.The result for each allele is a spatial correlogram, which is a plot of thecorrelation coefficients against distance. The correlograms for different allelesat a locus are not independent. In their applications, Sokal and his coworkersplot the correlograms for all but the least common allele at each locus.

This method does not combine information from different alleles. Everyallele is treated separately, although for convenience, the correlograms fordifferent alleles are plotted on the same graph. This method also does not yieldcorrelations for particular pairs of locations, but instead for all pairs of locationsseparated by the same distances. In fact, for a particular pair of locations and aparticular allele, a correlation coefficient does not exist because the only dataare the two allele frequencies. To compute a correlation coefficient, there mustbe an average taken over a set of pairs of values.

Sokal and his coworkers have used these methods to describe spatial patternsbut not to estimate parameters in population genetic models. They have shownthat hypothetical data with regular patterns in gene frequency and data gener-ated by Rohlf & Schnell's (105) model of isolation by distance will result incorrelograms that are significantly different from zero (118, 119). Sokal &Oden (118) summarize different mechanisms that can produce different kindsof correlograms. For example, they argue that positive autocorrelation over ashort distance might be due to migration (but not necessarily gene flow) over thesame distance or to natural selection acting in small patches. Similarities anddifferences in correlograms of different alleles could be used to detect similari-ties and differences in causal agents. For example, different correlograms atdifferent loci could indicate patterns of natural selection. Similar correlogramsat different loci could indicate a common cause such as migration.

The advantage of using these methods is that they require no assumptionsabout the history of the population or the mechanisms governing geneticevolution, because the correlograms are simply descriptions of the original

data. That, however, is also a weakness of the method. Without any assump-tions about the population, it . is impossible to estimate the strengths of forcesaffecting the population being studied. All that is possible is a qualitative comparison of correlograms, which can suggest possible causes of similaritiesand differences. To say anything more requires a population genetic or de-mographic model to predict what range of correlograms arc and are notconsistent with particular assumptions. Sokal & Wartenberg (119) computedspatial autocorrelations for data generated by a model of gene flow and geneticdrift, but they did not address this issue. They showed that significant correlo-grams are found in their model, but they did not examine what range ofdispersal distances is consistent with a particular correlogram. It is notable thatthey could not detect differences in correlograms for models in which thedispersal distances differed by a factor of two (i.e. between sets 3 and 4 in theirsimulations) (119).

Sokal & Wartenberg (119 : 220) state that the relationship between spatialautocorrelations and other measures of genetic differentiation, such as geneticcorrelations, gene identities, and F-statistics, has not been established. It is not.however, difficult to do so for neutral loci in a population at equilibrium. lithepopulation occupies an area much larger than the characteristic length associ-ated with gene flow and genetic drift—V(m/pL) in a stepping-stone model oro-/Vp, in a continuum model—then the correlogram for every allele is predictedby the theory of Kimura & Weiss (58) and MalecOt (72). To a good approxima-tion, the correlation in allele frequency depends only on the distance betweenthe populations, so the expected correlation obtained by averaging over allsubpopulations separated by the same distance is given by their formulae. Infact a more direct comparison with the analytic theory could be made by takingthe Fourier transforms of the correlograms, thereby obtaining power spectra.because the theory is most easily developed in terms of the Fourier transforms,as shown by Malecot (72).

One potential difficulty with using spatial autocorrelations to estimate in or

J

other parameters is that at equilibrium the extent of correlation with distancedepends on the ratio neti. Even under the assumption of complete neutrality,there is no reason to expect the same correlograms from different loci inpopulations at equilibrium, because mutation rates vary among loci.

Genetic Distances

Genetic distances were first used in population genetics to provide a singlequantitative measure of differences in two or more sets of allele frequencies(e.g. 3). Cavalli-Sforza & Edwards (13) were among the first to use geneticdistances to infer phylogenetic relationships among noninterbreeding pop-ulations. If a genetic distance between noninterbreeding populations increasesmonotonically with time, then the genetic distance between a pair of pop-


ulations indicates the time since they descended from a common ancestralpopulation. Felsenstein (35) reviews the problems and methods in relatinggenetic distances to phylogenies.

If there is continuing gene flow among populations, the genetic distance /between a pair of populations presumably decreases as the gene flow betweenthem increases. This is true but does not easily lead to useful conclusions aboutgene flow in nature. As previously discussed, two kinds of genetic distances areused, those such as Cavalli-Sforza & Edwards's (13) that have propertiessimilar to FsT because they depend on the differences among populations inheterozygosity, and those such as Nei's (93) that depend on differences inhomozygosities. This distinction was made by Latter (61) but has been largelyignored since. For estimating levels of gene flow, there seems no reason to usegenetic distances that depend on differences in heterozygosities instead of usingFsT directly. As indicated above, the statistical properties of Fs7- and itsrelationship to population genetic models are reasonably well known.

Nei (93) defined a measure of genetic distance and suggested that it can beused to estimate m. Nei's measure is defined to be D = -1n(JxylVJxJy), whereJx = /pj2 ,Jy=1 qi2 , Jxy = Epi qj , andp, and q, are the frequencies of alleles ata locus in populations X and Y. Its value depends on differences inhomozygosities within and between subpopulations and so is the second kind ofgenetic distance. In the n-island model, the expected values of both ./x and ./yare fo and the expected value of .1 xy is f, so Equation 2 implies:

D = In {1 + np.(1 - m) 2/[m(1 - m12)1}

11.

if the J's all take their expected values. If m and are both small, then Ddepends primarily on mIt.L. A similar dependence of D on m/p, is found in theone- and two-dimensional stepping-stone models (93).

Nei (95: 194) suggests estimating in from the relationship exp(-D) ---m/(m+ p.), which follows from Equation 11 when both in and are small, byassuming p = 2 x 10 -6 . For example, in three species of Drosophila, theaverage values ofD among populations are 0.008 for D. willistoni, 0.049 for D.tropicalis, and 0.043 for D. equinoxialis (1). The resulting estimates of m are2.5 X 10 -4 , 4.0x 10 -5 , and 4.5 x 10 -5 . These values are consistent with theestimates of Nm using both FsT and rare alleles found in Drosophila in generaland in D. willistoni in particular (114) if, as is reasonable to expect, theeffective populations sizes in these Drosophila species are 10 8 or larger. Insalamanders, estimates of m made using Nei's genetic distance are essentiallythe same as those for Drosophila (59, Table 2). The average for 24 plethodontidspecies is 2x 10 -5 , with the largest and smallest values being 2x 10 -4 and2 x 10 -6 . For those species, estimated population sizes are thought to be muchsmaller; they thereby also provide estimates of Nm consistent with estimatesusing rare alleles and FsT. Nonetheless, it is interesting that average values of Din such different species are so similar.

The difficulty with this method for estimating in is that it depends on severalassumptions, and the sensitivity of the estimates to changes in those assump-tions is largely unknown. The effect of differences in mutation rate among locicould probably be dealt with by assuming a reasonable distribution of mutationrates among loci, as has been done by Nei et al (96) for noninterbreedingpopulations. Selection appears to cause much more serious problems. Inunpublished simulations I have found that equilibrium values of D depend on1,!

III selection if the average selection coefficient is as large as or larger than tn.Oiyi Because estimates of m presented above are on the order of 10 -4 or smaller.) very weak selection would invalidate using D to estimate tn.

Genetic distances potentially provide a different kind of information about asubdivided population because a genetic distance between all pairs of sub-populations can be computed. From the matrix of genetic distances it might hepossible to estimate elements of the migration matrix or at least provide moreinformation about the elements of the migration matrix than simply the averagelevel of immigration, m. The importance of this problem has been discussed byFelsenstein (34). At present, there is no formal theory for estimating elementsof the migration matrix from the matrix of genetic distances. Maruyaina and I(115) simulated a two-dimensional stepping-stone model on a torus and showedthat there is only a weak relationship between geographic distance and Nei'sgenetic distances, probably too weak to have any statistical power.

Although formal justification is lacking, populations that have smaller genet-c distances between them are assumed to have higher levels of gene flowamong them than populations that have larger genetic distances. For example,in Nevo & Yang's (98) study of flyla arborea, they found that 7 of the 8populations sampled had very small values of D among them-0.004-0.67-but the values of D between those populations and one other were muchlarger-0.127-0.141 (98, Table 5). This result suggests that the one populationis relatively isolated. Assuming neutrality of these loci and ignoring samplingconsiderations, these results imply that the isolated population has a rate of geneflow with the other populations an order of magnitude lower than thosepopulations do among themselves. My analysis of the same data reachesessentially the same conclusion (114). With both methods, the statisticalproblems associated with estimating the degree of isolation are unexplored.Because observed genetic distances within species are usually small, there maybe a large uncertainty in estimates of /11 for a particular pair of populations, evenassuming neutrality, equal mutation rates, and large sample sizes.

Rates of Approach to Equilibrium Conditions

C- All indirect methods discussed above assume that populations being studied arein a genetic equilibrium. This is probably not true for many species and iscertainly not true for most human populations. For both FsT and the conditional

- average frequencies, the rate of approach to an equilibrium in the n-island

rI!

414 SLATKIN GENE FLOW 4 15

model is on a time scale of the same order of magnitude as 1/m (17, 114). Thisbehavior is somewhat surprising because the time scale of approach to theoverall genetic equilibrium is of the order of magnitude of 1/p. (87), which isprobably much larger than 1/m. Both FsT and the conditional average frequen-cies combine information on allele frequencies in ways that mask the fact thatthe entire population may not be at a genetic equilibrium. Thompson (123) hasdeveloped a method for using the distribution of rare alleles to infer the times ofpopulation fission, and she has applied that method to data from humanpopulations; her method, however, seems impractical for events occurringmore than a few generations in the past.

For Nei's genetic distance, Nei & Feldman (97) found that the rate ofapproach to the equilibrium depended on the larger of 1/m and lip, which ) underthe assumption that m>> 11, implies that Nei's genetic distance takes muchlonger to reach its equilibrium value than FsT or the conditional averagefrequencies. If there are low levels of gene flow, all of these measures take along time to reach an equilibrium.

For spatial autocorrelation methods to indicate the pattern of gene flow, thepopulation to which they are applied must also be at an equilibrium. The overallequilibrium is reached at a time comparable to lip., but it is not known at presentwhether the patterns in the spatial autocorrelations do approach an equilibriummore rapidly. Given the similarity of correlations in gene frequencies to Nei'smeasure of genetic distance, that is probably not the case. It is possible thatspatial autocorrelations are useful for detecting very recent gene flow andpopulation dispersal, as has been suggested by Sokal (117), but as yet there isno theoretical justification and no way to quantify the amount of gene flowcausing the observed patterns.

DIRECT MEASUREMENTS OF DISPERSALAND GENE FLOW

The potential for gene flow due to the dispersal of individuals at particularlife-history stages can be assessed by direct observation of those stages or by therelease and recapture of marked individuals. Actual gene flow can be measureddirectly by examining the spread of distinctive alleles in a population. Thesemethods are often preferred to indirect methods because the former appear torequire fewer and less restrictive assumptions. In fact, direct measures of geneflow require different assumptions and yield a different kind of informationthan indirect measures. Because of the limited time scale of observation andexperimentation, direct measures cannot detect occasional and unpredictablechanges in the level and pattern of gene flow, even though the theoriesdescribed above show that the evolutionary effect of gene flow is greatlenhanced by a few long-distance migrants and by the extinction and recoloniza- GcIL

tion of local populations. Direct measurements indicate the gene flow occurringunder the conditions when the measurements were made, rather than theaverage level of gene flow. Observations and experiments tend to be takenunder normal or natural environmental conditions, yet exceptional conditionsmay cause unusually high yet evolutionarily important movements of genes.

Dispersal of Marked Individuals

Direct observations of animal dispersal have been done for many reasons.Probably the first such study to answer evolutionary questions is that byDobzhansky & Wright (23) on Drosophila pseudoobscura. Dobzhansky &Wright released individuals homozygous for the orange mutant and collectedflies in traps 10 or 20 meters apart laid out in a crossed pattern. The trapscontained a banana-yeast mixture attractive to flies. The researchers verifiedthat the orange allele did not affect dispersal when they observed similardispersal in individuals marked with platinum nail polish. DobzhanskyWright found that the standard deviation (cr) of the dispersal distance at 22°Cwas a total of 200-250 m, with roughly half of the dispersal occurring in the firstday and with a maximum distance of 500 m. Dispersal was slower at lowertemperatures, and no dispersal occurred at temperatures below I 5°C. Using anestimate of the average density of 0.005 m -2 , they estimated the neighborhoodsize to be 500-1000 individuals, a value consistent with researchers' previousestimates of effective population size from the allelism of lethals (22). Theconclusion, which has been widely cited, is that gene flow in D. pseudoobseura

is restricted. Even though the species occupies a range of millions of hectares,individuals separated by as much as a few kilometers are effectively in in-dependent populations.

Other studies of dispersal in Drosophila, particularly pseudoobscura, but

also of melanogaster (125), nigrospiracula (52), aldrichi (103), willi.vtoni ( I 1),

and other species, have used similar methods and largely have confirmedDobzhansky & Wright's conclusion that dispersal is highly localized. The mostdetailed study is that of Crumpacker & Williams (20) on D. pseudoobscura inColorado. They performed a careful statistical analysis of their data and foundaverage dispersal distances to be about 50% higher than in Dobzhansky &Wright's study. Crumpacker & Williams estimated the neighborhood size to beapproximately 10,000, almost an order of magnitude larger than Dobzhansky &Wright's estimate.

Dobzhansky et al (21) found that, for D. pseudoobscura in the same locale

where the original Dobzhansky & Wright experiments were done, the averagedispersal distances were roughly the same as in the earlier study but thatdispersal distances in different habitat types differed by almost an order ofmagnitude. Flies dispersed least in dense woods, which was also the habitatwith the highest population densities, and dispersed most in meadows and open

416 SLATKIN

GENE FLOW 417

woods, which had the lowest population densities. As Dobzhansky et al (21)indicate, these results have the obvious interpretation that flies disperse lesswhen they are in favorable habitats than when they are in unfavorable ones,which implies that dispersal is not a species-specific behavior but a behaviorthat varies with environmental conditions.

Johnston & Heed (51) have questioned the utility of banana-yeast baits forestimating dispersal distances. They point out that the baits are such attractivefood resources that recapture data depend on foraging behavior more thandispersal behavior. In their study of D. nigrospiracula, their estimate ofaverage dispersal distance using baits was much smaller than the averagespacing of necrotic saguaro cacti, the only known natural habitat of this species.They also reanalyzed data from other studies of dispersal in Drosophila andshowed that estimates of average dispersal distance are highly correlated withthe spacing of baits in those studies. Although no one has done so, it would bepossible to use the model developed by Bossert & Wilson (9) to find the "activezone" of a bait, the volume within which an individual would experienceconcentrations above a specified threshold level. Crumpacker & Williamsestimated the "attractive radius" of their traps to be 46 m; this was based on howfar apart traps had to be to not affect each other. Johnston & Heed's results donot refute Dobzhansky et al's conclusion (21) that dispersal distances depend onhabitat, because in their study dispersal distances were estimated using thesame spacing of baits in every habitat.

More recent studies of D. pseudoobscura have confirmed that dispersalbehavior is a manifestation of foraging behavior and habitat selection. Jones etal (53) studied populations in Death Valley, in which populations exist only atoases that are separated by several kilometers. They found the average dispersaldistance after one day to be 400-500 m, roughly three times that found byDobzhansky & Wright (23). Jones et al released flies in clearly unsuitablehabitats in the desert and recaptured some the next day not only at the nearestoasis 2 km from the release point but at a farther oasis 8 km away. Coyne et al(16) showed that long-distance dispersal occurred even when flies were re-leased in an oasis. Most flies released in an oasis were recaptured in the sameoasis, but marked flies were recaptured as far as 5 km away in the desert and atan oasis 14.6 km away.

There have been numerous studies of the dispersal of marked individuals, butI will not attempt to review them here. Endler (29) provides a useful summaryof this literature up to 1976. The conclusion from these studies is that dispersalin most species is very confined. Even though there is the physical capacity tomove long distances and the literature contains many reports of individualsbeing found exceptionally far from the nearest possible source population,individual organisms do not tend to move nearly as far as they are able.

The checkerspot butterfly, Euphydryas editha, is another example widely

cited as providing evidence of restricted dispersal. Ehrlich (25) found that ofover 1000 adults released in three areas of Jasper Ridge, more than 975- wererecaptured in the same area, even though the total length of the habitat suitablefor the butterflies was about 200 m. Ehrlich et al (28) discuss the question ofwhy even the few individuals that did change areas probably did not breedsuccessfully. Gilbert & Singer (38) studied a different population of E. editha,which is of a different ecotype, and found much larger dispersal distances. Theyconcluded that dispersal tendencies depend on habitat, and this agrees with thelater results from Drosophila. They also showed that in the Jasper Ridgepopulations, dispersal was negatively correlated with population density.suggesting that high densities were indicators of high quality habitats. Differentecotypes of E. editha have times of breeding during the year sufficientlydifferent that gene flow among permanent populations of different ecotypesseems precluded (28).

For plethodontid salamanders, Larson et al (59) have thoroughly reviewedavailable data. They conclude that long-distance gene flow is unlikely bothbecause observations of behavior show the salamanders have low dispersaltendencies and because many populations are separated by habitat completelyunsuitable for salamanders.

For species, including most birds and mammals, in which social behaviorinfluences mating, there seem to be two opinions: one is that social behaviorreduces the amount of gene flow and the other is that it promotes gene flow. Thefirst view is exemplified by the discussion of Selander et al (108), who arguethat the extreme differentiation of gene frequencies of Peromvscus is due in partto a tendency to interbreed within family groups. Individuals unknown tomembers of a group are attacked. In mice in particular, the view that there isextreme isolation of social groups is supported by the hypothesis that a reces-sive lethal allele at the T-locus is kept in low frequency by interdemic selection.which implies that there is little gene flow (2).

The contrary opinion—that social behavior promotes gene flow—is ex-pressed by Schwartz & Armitage (106), who argue that the social system ofmarmots forces males to disperse from their natal group. Both of these viewsimply that dispersal is strongly affected by local population densities. Endler(29, Table 6.1) compiles several estimates of dispersal distances in mammals(and other species). For mammals, estimates of o-, the standard deviation ofdispersal distances, are between 10 and 100 m; this is based mostly on data fromrodents. In birds, dispersal distances vary greatly, with little apparent correla-tion with breeding system (40).

Another group for which dispersal is well studied is snails, particularly

Cepaea nemoralis and its sibling species C. hortensis. Individuals move a netdistance of a few meters per year, and estimates of o- are from 30 to 50 m (54).How strongly dispersal depends on habitat is not known. These opportunistic


species rapidly invade suitable habitats; what they lack in speed they apparently

make up for in persistence.In plants, dispersal of pollen and seeds cannot be observed but can be inferred

from the properties of pollinating and dispersing agents. Levin & Kerster (66)provide an excellent review of this literature. The average distances traveled byinsect pollinators, particulary honeybees and butterflies, are generally short andthe distribution is often extremely leptokurtotic. Levin & Kerster (64) found thedistribution of flight distances had a mean of 0.58 m and a kurtosis of 26.11. Asin the case of dispersal by animals, pollen dispersal is a product of foragingbehavior. Levin & Kerster (65) also found that honeybees traveled longerdistances when plants were in lower densities. For plants pollinated by hum-mingbirds, territorial behavior tends to restrict pollen flow (66).

Not all animal-mediated pollen dispersal is over short distances. Janzen (50)showed that eugolossine bees travel hundreds of meters or farther betweenflowers. Even that may not cause extensive gene flow because those beesvisited plants that are probably each other's nearest neighbors. True long-distance gene flow by animal pollinators would require either than an animalvisit large numbers of plants and carry pollen from those visited early to thosevisited much later or that an animal skip numerous suitable plants betweenvisits. The first possibility seems unlikely because it would require that pollenfrom plants visited early in a foraging bout be saved until later in the bout.While it is known that not all pollen is released at the first plant visited, most ofit is released at the first few plants. The second possibility seems unlikelybecause it is inconsistent with what is known about foraging behavior ofanimals. I agree with Levin & Kerster (66) that pollination by animals isunlikely to result in long-distance gene flow.

Wind-dispersed pollen can go much farther than pollen dispersed by animals,and the distribution of dispersal distances is affected by population density onlyto the extent that density influences air currents. Although pollen can be carriedlong distances by winds, it seems unlikely that wind-dispersal of pollen resultsin long-distance gene flow, because of the low probability that it would land inflowers of the right species. For both wind-dispersed and animal-dispersed jpollen, the distribution of dispersal distances probably gives an underestimateof the actual extent of gene flow. If neighboring individuals are closely related,pollen coming from nearby individuals can be significantly less successful thanpollen coming from farther away, both because of inbreeding depression andbecause of self-sterility systems. Levin (62) has recently reviewed this topicand concluded that, even when pollen dispersal appears to be highly localized,effective gene flow is occurring over somewhat longer distances. The effect is,however, a quantitative one and does not seem to result in a qualitativelydifferent pattern of dispersal.

Seed dispersal is potentially a much greater source of gene flow in plants, but

there are no data on long-distance seed dispersal. Estimated dispersal distancesof seeds are similar to ihose for pollen dispersal. For example, Boyer (10)examined seed dispersal in Pinus palustris near the edge of a stand and foundmost seeds recovered in traps were found within 27 In, and few were beyond100 m. Harper (45) reviewed numerous other studies of seed dispersal andfound that dispersal distances vary from a few meters to a few kilometers. Otherinformation, though, suggests that seed dispersal may commonly result inlong-distance gene flow. Effective long-distance dispersal of seeds is essentialfor succession in plant communities. The speed at which many plant speciescolonize vacant habitats implies that gene flow between established populationsdue to seed dispersal may be common (63). Furthermore, in some plants.characteristics of seeds and fruits make them attractive and accessible toanimals (e.g. 49). Whether or not these characteristics were favored by naturalselection to promote dispersal is unknown, but they certainly facilitate theconsumption of seeds by animals capable of carrying them long distances anddepositing them in other suitable habitats.

The general conclusion from observations of dispersal is that in many andperhaps most species dispersal is highly localized, although obvious exceptionssuch as marine species with planktonic larvae are known. This may not implythat gene flow in natural populations is necessarily so restricted. Because of themethods used in these studies, it is impossible to estimate the frequency oflong-distance dispersal that occurs under normal conditions or in response toextreme changes in the environment. In addition, these direct methods do notestimate the importance of extinctions and recolonizations as a source of geneflow.

Dispersal of Marker Alleles

Dobzhansky & Wright (23) used the orange allele as a nondisruptive marker ofreleased flies but did not monitor the frequency of the orange allele in thepopulation. In a later study (24) of D. pseudoobscura, they did record thefrequency of individuals heterozygous for the orange allele several weeks afterthe release and concluded that 95% of the progeny of the flies released werelocated within a circle of radius 1.76 km from the point of release. Distinctivealleles, often dominant alleles affecting flower color, have been used to exam-ine gene flow in plants, first by Bateman (5, 6), and more recently by Gleaves(39), Handel (43), and others. The results of these studies and others areconsistent with estimates of pollen dispersal: Gene flow tends to be confined tothe immediate neighborhood of an individual. The distributions of dispersal/distances do depend on the density of conspecifics in the surrounding area.Bateman, Gleaves, and Handel have also examined the details of the shape ofthe distribution of dispersal distances, concentrating particularly on the extentof kurtosis and isotropy. These and other properties of the distribution are

420 SLATKIN

GENE FLOW 421

important for understanding mechanisms causing dispersal but are of lessimportance to understanding the potential for local genetic differentiation inthose species. As indicated above, in a continuously distributed population, thevariance of dispersal distances, cr2 , determines both the potential for localdifferentiation and the length scale of correlation between different geographiclocations.

Jones et al (53) used marker alleles to determine whether populations of D.pseudoobscura in oases in Death Valley are permanent isolated populations.No individuals can be trapped during the summers, but the populations could beeither reestablished in the autumn by aestivating individuals from the previousspring or refounded by individuals from other populations. Jones et al (53)introduced individuals homozygous for an unusual allele at the esterase-5locus. The allele was in low frequency (2%) in the undisturbed populations.Thirty-thousand individuals homozygous for the allele were released at each oftwo oases, resulting in a frequency of 78% after 38 days. At the end of thespring, the frequency had dropped to 6% and when the populations reappeared,no copies of the allele were found, although only 43 individuals were captured.These results suggest but do not conclusively show that the oasis populationswere recolonized from permanent populations, the nearest of which is 15 kmaway.

Baker (2) introduced individuals carrying an unusual hemoglobin allele inMus musculus and found that it spread to neighboring populations within twoyears. Based on that observation and direct estimates of dispersal, Bakerconcluded that gene flow was too common for neighboring populations todiverge because of genetic drift, in contrast with previous opinions aboutdispersal in Mus.

Extinction and Recolonization of Populations

The extinction and recolonization of local populations is a form of gene flow,possibly a very important form. As indicated above, extinction and recoloniza-tion caneasily be more effective than dispersal between established populationsin preventing local differentiation. This type of gene flow is easy to overlookpartly because it falls in the realm of ecological rather then genetic studies andpartly because it may not be regarded as representing normal events. It has notcompletely escaped attention, however. Selander (107) argued that extinctionsand recolonizations could account for the patterns in allele frequencies in Helixsnails. Heed (46) pointed out that several Drosophila species are found on akipuka (an island of vegetation surrounded by lava) of recent origin, whichimplies rapid colonization.

Although it might be reasonable to expect that the dispersal tendencies inestablished populations would indicate the potential for colonizing vacanthabitats, that need not be the case. A good example is provided by Euphydryas

editha. As indicated above, the frequency of exchange between establishedpopulations is low. Yet on one occasion. when one of the populations wentextinct, a new population appeared in the following year, presumably made upof colonists from one of the two neighboring populations (85).

A similar picture emerges in relation to Drosophila, although recent studiescited above show dispersal may not be as restricted as previously thought.Moore et al (84) collected specimens in an area recently burned and foundindividuals of four species of Drosophila several hundred meters into theburned area only two weeks after the fire. This study was done in an area similarto that studied by Dobzhansky & Wright (23). Jones et al (53) suggest that therecolonization of populations in desert oases occurs every year, from pop-ulations at least 15 km away.

No careful studies of extinction and colonization rates under natural con-ditions exist, so it is impossible to draw any general conclusions about thismechanism of gene flow. MacArthur & Wilson (69) argue that extinctions andrecolonizations are common on islands and island-like habitats and are respon-sible for patterns of species diversity. That is not the same as showing thatextinctions and recolonizations cause significant genetic changes in each spe-cies, but it is consistent with that view.

There are at least two ways to estimate rates of extinction and recolonization.One is to monitor new areas of suitable habitat, either areas previously occupiedor newly created, as has often been done in the study of island biogeography.This method provides an estimate of rates of recolonization. A second way is toestimate the fraction of available habitats that are occupied by the species ofinterest at any time. Under relatively simple assumptions about the demographyof a species (69), the fraction of available habitats can he related to the ratio ofthe extinction and recolonization rates. This method assumes that suitablehabitats for a species can be identified, something that may be relatively easyfor some species, as it is for many fresh water fishes that are introduced as gamefishes in lakes and rivers; it is probably impossible for other species, for reasonsdiscussed by Rabinowitz (101).

DISCUSSION

In the previous sections, I have described several ways to estimate levels ofgene flow. I will discuss a few examples in which some of these methods havebeen used. In some cases, different methods lead to the same conclusion aboutgene flow, but in others indirect methods indicate much higher levels of geneflow than are suggested by direct methods.

DROSOPHILA Drosophila is one of the most thoroughly studied genera de-spite an almost complete lack of knowledge about the ecology of most species

422 SLATKIN

under natural conditions. Only the cactophilous and Hawaiian species seem tohave a life outside collecting traps. The extensive studies of D. pseudoobscuraespecially illustrate the difference in conclusions based on direct and indirectmethods. Both conditional average frequencies and FST values indicate that anaverage value of Wm of 1 or greater in an island model is needed to be consistentwith the electrophoretic data. The consistency of these two indirect methods isespecially significant because FST is most strongly affected by common, notrare, alleles. On the other hand, dispersal distances over the lifetime of in-dividuals in what is apparently the most favorable habitat are on the order ofhundreds of meters, but the geographic range of the species is much larger.Although individuals do fly much longer distances in deserts, desert habitatsmake up only a small fraction of the range and probably contain a much smallerfraction of the total population.

This seeming paradox can be resolved in one of several ways. Obviously oneway is to discount the indirect methods completely, either because the theoret-ical results are incorrect or the data are inadequate. I think this is not asuccessful resolution. The theory is sufficiently well understood that its quali-tative features, if not all its quantitative details, are correct. It is also likely thatpatterns in the data are real despite the known limitations of electrophoreticmethods. In Drosophila, as in other species for which indirect methods showhigh levels of gene flow, the application of two different indirect methodsyields similar results and the same pattern is exhibited by all or almost all loci. Itwould require remarkable bias in the data to exactly mimic a pattern of a highlevel of gene flow at most loci in a species in which there is actually a low level

of gene flow.Another possible resolution of this paradox, and the one I favor, is that the

disparity between the results of applying direct and indirect methods is real andindicates that gene flow in natural populations is highly stochastic. Directestimates of dispersal distances can be biased because individuals under normalconditions, when direct estimates of dispersal are usually made, do not movevery far. But under some conditions—after an extreme environmental change,after the arrival of a new predator or competitor, or with the appearance of avacant habitat—long-distance gene flow can and does occur. Even though suchevents may be relatively rare, they may be common enough to produce genetichomogeneity of different populations of a species such as D. pseudoobscurathat is distributed over a large geographic range.

EUPHYDRYAS The checkerspot butterfly, Euphydryas editha, is similar inmany ways to D. pseudoobscura. Direct observations of E. editha' s dispersalbehavior show even more restricted movements than in Drosophila despite theapparently greater capacity for long-distance flight. Ehrlich & Raven (27) used

E. editha as one of their most prominent examples in discussing the evolution-

GENE FLOW 423

ary role of gene flow. In other papers (28, 82), Ehrlich and his coworkers havereiterated that gene flow in E. editha is of little importance in determining genefrequencies.

As in the case of D. pseudoobscura, indirect methods indicate that gene flowis an important force in homogenizing local populations. McKechiiie et al (82.Table 1), present electrophoretic data for eight polymorphic loci from 21 sitesin California. They computed an estimate of F sT for each allele. Seven of theeight loci had similar average values of F sT in the range (0.016. 0.037) withone, hexokinase, having a much larger average value, 0.291. I obtained theseaverage values by weighting the published FsT values for each allele at a locusequally. The average F sT over all loci is 0.0638 which gives an estimate of Nmof 3.7. If hexokinase is ignored, the estimate of N171 increases to 7.7. Becausethere is only one private allele (whose frequency is 0.01), I could not obtain aquantitative estimate of Nm using my method, but the conditional averagefrequencies of alleles are consistent with Nm being somewhat larger than one.

McKechnie et al (82) argue that their data show evidence of selection; theyuse the argument that the low levels of gene flow indicated by their directobservations of dispersal require selection to account for the similarity of allelefrequencies in different locations. If there is indeed no or little gene flow, thenthe data do show evidence of selection. I analyzed their data using my methodfor testing selective neutrality in a subdivided population that has low levels ofgene flow (l 13). My method is to apply the Ewens's (30) test to data from eachlocation separately and then combine tests for different locations to gainstatistical power. All eight loci show evidence of selection, with six havingsignificantly more homozygosity than expected and two having significantlyless. Although there is evidence for selection (under the assumption of no geneflow) the selection on most of the loci is not in favor of heterozygotes, asclaimed by McKechnie et al, because that type of selection would result inexcess heterozygosity. If there is indeed little or no gene flow, then differentkinds of selection are affecting different loci, yet producing patterns in allelefrequencies at those loci that are similar and that indicate high levels of geneflow.

An alternative explanation is that the indirect methods are revealing theactual high level of gene flow in this species. That level of gene flow wouldinvalidate the application of my method for testing neutrality because thatmethod assumes samples are taken from independent populations. The appar-ent selection detected by applying my test of neutrality can be explained by thecorrelations among local populations caused by gene flow among them. Thisexplanation also accounts for the fact that McKechnie et al (82: 586) say theyfound no evidence of selection when they applied Ewens's test to the data fromall the populations together. That would be expected if the loci are neutral and

there is a high level of gene flow (113).

424 SLATKINGENE FLOW 425

In other studies, Ehrlich and his coworkers have found evidence suggestingthat the extinction of local populations is frequent. As indicated earlier, one ofthe populations (JRG) on Jasper Ridge went extinct twice in 15 years. After thefirst extinction it was recolonized in the next year. That population thenpersisted for two more years, went extinct, and has not since been recolonized(85). The failure of a second recolonization is almost certainly due to apermanent change in the plant community rather than a lack of availablecolonists (D. D. Murphy, personal communication). Other studies foundnumerous local extinctions caused by unusually dry conditions from 1975 to1977 (26), but at present none of those sites has been naturally recolonizeddespite an apparent recovery of potential host plants (D. D. Murphy, personalcommunication). Frequent extinctions and recolonizations could account forthe difference between observations of dispersal and patterns in gene frequen-cies, but much more has to be learned about the recolonization process before

any firm conclusions can be drawn.

SALAMANDERS In many other species, direct and indirect methods lead tothe same conclusions. Larson et al (59) review data from plethodontidsalamanders and conclude that, for many salamander species, dispersal doesnot occur among local populations. Both FsT values and conditional average

frequencies yield estimates of Nm much less than one. Although Larson et al(59) use a method, different from the one I subsequently developed (114), forobtaining a quantitative estimate of Nm from rare allele data, their estimates areof the correct order of magnitude and are generally consistent with estimates

obtained from FsT values. Both indirect methods show low levels of gene flow,and that is consistent with the direct estimates. Larson et al (59) emphasize thedifference between the direct and indirect methods and conclude that theindirect methods are deficient. I do not agree because both indirect methods are

not likely to be accurate when Nm<<1. For that range of parameter values, anequilibrium is not reached for a long time, and the equilibrium distributions ofgene frequencies when there are low levels of gene flow are sensitive to slightamounts of selection. For either indirect method to indicate that Nm = 0,different alleles would have to be found in every location and at every locus, asituation that has not been found within any species or even when comparingdifferent species. I also disagree with the conclusion of Larson et al (59) thatindirect methods can provide useful information about the demographic historyof a species in which there is currently no gene flow. In my simulations, I have

found that the patterns in both rare alleles and F57 when there is no gene flow aretoo sensitive to weak selection to be of use in understanding the history ofpopulations. In any case, the evolutionary implications of continued low levelsof gene flow and no current gene flow are the same: Gene flow is not strongenough to cause homogeneity even at neutral loci.

There are many other species for which direct and indirect methods makeconsistent predictions. The mussel, Mvtilus edulis appears to have very highlevels of gene flow over a range from Nova Scotia to Long Island (112, 114),which is consistent with its having planktonic larvae. On the other hand. thesnails Cepaea nemoralis and Helix aspersa show significant local differentia-tion in allele and phenotype frequencies over short distances (54, 107). which isconsistent with their relative immobility. Snails do colonize new areas, andSelander (107) has shown that although sampling during recolonization may beimportant in Helix, extinctions and recolonizations are not frequent enough toresult in significant genetic homogenization even among populations separatedby only tens of meters.

CONCLUSIONS

There is still much to be learned about gene flow in natural populations, but wecan reach some tentative conclusions. (a) There are useful indirect methods forestimating the average level of gene flow from the spatial distribution of allelefrequencies. (b) These methods show average levels of gene flow differ greatlyamong species. (c) Differences in levels of gene flow cannot account formorphological stasis. Salamanders, which have low levels of gene flow, areamong the oldest extant terrestrial vertebrates and are some of the mostmorphologically homogeneous (124). (d) Gene flow in some species appears tobe sporadic, with dispersal over short distances occurring most of the time butwith dispersal over much longer distances occurring frequently enough toproduce widespread genetic homogeneity.

A question that cannot yet be answered is whether gene flow at whateverlevel it occurs plays primarily a conservative or a creative role. The differencebetween Mayr's (80) and Ehrlich & Raven's (27) views of the importance ofgene flow concerns its conservative role: Does gene flow prevent adaptation tolocal conditions? In Wright's "shifting balance theory," (see 135 for the mostrecent summary), gene flow also plays a creative role. If there are low levels ofgene flow, well-adapted combinations of genes can be fixed in one localpopulation through the combined action of genetic drift and natural selectionand then spread to other local populations by gene flow. In Wright's theory bothdispersal between established populations and extinctions and recolonizations(augmented by interdemic selection) are important for spreading new com-binations of genes. Species in which direct and indirect methods indicatedifferent levels of gene flow seem good candidates for testing Wright's theory.Low levels of dispersal under most conditions could provide the opportunity forwell-adapted combinations of genes to be fixed, and occasional episodes ofcolonization and extinction could spread those combinations to other pop-ulations. Species in which direct and indirect methods are consistent seem

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45. Harper, J. L. 1977. Population Biologyof Plants. New York: Academic

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47. Huxley, J. S. 1938. Clines: an auxilliarymethod in taxonomy. Nature 142:219-20

48. Jackson, J. F., Pounds, J. A. 1979. Com-ments on assessing the dedifferentiatingeffect of gene flow. Srst. "Loo!. 28:78-85

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426 SLATKIN

poorer candidates. If levels of gene flow are always high, well-adapted com-binations of genes cannot become established in any population, and if levels ofgene flow are always low, those combinations could be established in onepopulation but could not spread to other populations.

Both the data and theory need refining before we can understand more aboutthe evolutionary role of gene flow. Data on geographic variation in restrictionsite polymorphisms and in DNA sequences would be more directly related tothe mathematical models than are electrophoretic data. Theory is needed thatcan reveal transient patterns in allele frequencies caused by sporadic gene flowand that can reveal the actual pattern of gene flow rather than an overall averagelevel. Detailed observations of dispersal are needed in more species for whichadequate genetic data can be obtained, with particular attention paid to dispersalover long distances and under unusual environmental conditions.

ACKNOWLEDGMENTS

This review was supported by NSF grant DEB-8120580. I thank J. Felsensteinfor extensive and patient discussion of much of the material in this paper and S.Adolph, P. R. Ehrlich, J. Felsenstein, A. B. Harper, D. D. Murphy, R.Olmstead, and R. Shaw for numerous helpful comments on the first draft of this

paper.

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Ann. Rev. Ecol. Syst. 1985. 16:431-46Copyright © 1985 by Annual Reviews Mc. All rights reserved

COMPATIBILITY METHODS INSYSTEMATICS

Christopher A. Meacham*

Botany Department, University of Georgia, Athens, Georgia 30602

George F. Estabrook

Department of Botany, University of Michigan, Ann Arbor, Michigan 48109

THE BASIC IDEA

Character compatibility analysis is founded on the idea that, for the purpose ofmaking plausible reconstructions of evolutionary relationships among the spe-cies or other evolutionary units in a taxon, characters are already hypotheses ofevolutionary relationships. If two or more such hypotheses are logically con-sistent (compatible), then they may be combined into a single, more complex,hypothesis that often asserts a more refined, resolved reconstruction of evolu-tionary relationships. If two such hypotheses are logically inconsistent (in-compatible), they cannot be combined without first modifying one or the otheror both so that they become compatible. When used to construct a phylogenetictree, algorithms (such as those based on minimizing evolutionary changes) that

can accept incompatible characters for input make these modifications auto-matically without additional scientific considerations. Knowledge of the waystwo incompatible characters contradict one another as hypotheses can be usefulwhen one considers which possible additional research activities might suggesthow the hypotheses should be modified. Collections of mutually compatiblecharacters can help to determine which characters plausibly reflect true histori-cal evolutionary relationships.

To establish a rigorous analytic technology for character compatibility anal-ysis, we must make precise the notion of a character as a hypothesis of

*Current address is: University Herbarium, University of California, Berkeley, CA 94720.

4310066-4162/85/1120-0431$02.00

gene flow in natural populations es. izt3- 31s-

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