Abstract
The subdivision of a species into local populations causes its response to selection to change, even if selection is uniform across space. Population structure increases the frequency of homozygotes and therefore makes selection on homozygous effects more effective. However, population subdivision can increase the probability of competition among relatives, which may reduce the efficacy of selection. As a result, the response to selection can be either increased or decreased in a subdivided population relative to an undivided one, depending on the dominance coefficient F_{ST} and whether selection is hard or soft. Realistic levels of population structure tend to reduce the mean frequency of deleterious alleles. The mutation load tends to be decreased in a subdivided population for recessive alleles, as does the expected inbreeding depression. The magnitude of the effects of population subdivision tends to be greatest in species with hard selection rather than soft selection. Population structure can play an important role in determining the mean fitness of populations at equilibrium between mutation and selection.
THE subdivision of species into local populations has been much studied, typically from the point of view of the differentiation of neutral allele frequencies or the adaptation of these demes to locally divergent conditions (Felsenstein 1976; Hedricket al. 1976; Hedrick 1986; Barton 2001). Yet most loci must be under some selection, and arguably most selection must be largely independent of local conditions, because deleterious, lossoffunction mutations at most loci are likely to decrease fitness over a broad range of circumstances. For these classes of uniformly selected mutations, however, we have little population genetic theory appropriate for subdivided populations.
The subdivision of a species into spatially isolated populations affects the outcome of selection in several ways. Population structure engenders nonrandom mating, because organisms are more likely to mate with nearby individuals than those far away. This results in an excess of homozygotes relative to that expected under random mating. If an uncommon allele is less than completely dominant to a common allele, then this excess of homozygotes will allow a greater response to selection at this locus. More generally, with any deviation from additive gene action, the marginal effects of alleles will change as a function of their likelihood of expression as homozygotes. The effects of inbreeding within a population on the response to selection have been extensively studied (Wright 1942; Ohta and Cockerham 1974; Lande and Schemske 1985; Charlesworth and Charlesworth 1987; Caballero and Hill 1992; Pollak and Sabran 1992; Pollak 1995), but population structure introduces new complications. In particular, if alleles are clustered in space and if the absolute success of individuals depends on some locally limited resource, then the success of one individual will disproportionately affect the reproductive success of other individuals carrying similar alleles. Another way of saying this is that the genetic variance within a population tends to be reduced by local drift, such that the response to selection is lowered. Thus with population structure, the response to selection may be increased by the greater expression of homozygotes but decreased by the effects of local drift and local competition. Barton and Whitlock (1997) briefly discussed the effects of population subdivision with soft selection on a locus with additive effects and found that the effects of population structure were relatively minor in this case. Here this model is extended to include arbitrary dominance and a broader range of population structures and modes of selection. Simple expressions for the response to selection, mutation load, and the inbreeding depression in structured populations are found. Population structure can have a large effect on these important quantities.
CHANGE IN ALLELE FREQUENCY BY SELECTION
Definitions and moments of the gene frequency distribution among populations: Consider the case of a locus with two alleles, one fit and one somewhat deleterious. The frequencies of these alleles within deme i are given by p_{i} and q_{i}, respectively. For diploid individuals, the relative fitnesses of the three possible genotypes are given by 1, 1 + hs, and 1 + s, respectively. Assume for now that there is random mating within each deme such that the genotypes are present in local HardyWeinberg proportions. In this case, the local mean relative fitness is given by
The overall change in the allele frequency of the metapopulation depends on the relationship between the mean fitness of a local population and its contribution to the next generation. Two extreme possibilities are typically considered: soft and hard selection (Wallace 1968). With soft selection, each deme contributes to the next generation independently of its mean relative fitness, whereas with hard selection, its contribution is proportional to its mean relative fitness. It is possible to scale between these two extremes; we can scale the relationship between the genetic makeup of a deme and its contribution to the next generation by a linear function with slope b. Define N_{i} as the size of deme i, N_{tot} = ΣN_{i}, and
Note that
The mean relative fitness of all individuals in the population can be calculated as
Hard selection: With hard selection, demes are represented in the next generation in proportion to their average fitness. If we assume, as we do for the rest of this article, that the nongenetic determinants of the contribution of a deme (C_{i}) are not correlated with allele frequency, then the expected change due to selection in the overall allele frequency is given by
Pure hard selection corresponds exactly to the case of inbreeding within an undivided population, as treated previously (Caballeroet al. 1991), with F_{ST} used in place of the inbreeding coefficient in these equations.
Soft selection: The allele frequency will change over one generation within a population as a result of soft selection as
Generalizing the hardsoft dichotomy: The overall change in allele frequency is
Limitations: These approximations have made a few assumptions, and their easy use requires a few more assumptions. Relatively standard assumptions have been made about the strength of selection, in particular that s ⪡ 1. To use the assumption that the neutral expectation of F_{ST} is sufficient to describe the F_{ST} of selected loci, then it must also be true that N_{i} s < 1. However, for the deterministic equations given in this article to suffice, it must be the case that the allele is not nearly neutral at the species level; i.e., N_{tot} s > 1. Together these assumptions will require that the number of demes in the species is not small. It is likely that the violation of the weak selection assumption will not cause a qualitative change in the conclusions below, but certainly there will be quantitative deviations from the predictions as selection gets strong.
Perhaps most importantly, it has been assumed that the strength of selection is equal everywhere. Clearly, there is an important class of mutations that will vary not only in the magnitude but also in the direction of selection in different subpopulations. This variability in what is locally adaptive will clearly change the expectations of mutation load and inbreeding depression from those derived later in this article. Relaxing these assumptions represents a major challenge for future work.
MUTATIONSELECTION BALANCE
Now let us focus on deleterious mutations, such that s < 0. With weak mutation, the change in average deleterious allele frequency from one generation to another is given by
The frequency of a deleterious allele at equilibrium is likely to be much smaller in a subdivided population than in a panmictic population (see Figure 1). Selection is more effective in subdivided populations (i.e.,
MUTATION LOAD
The previous section has shown that under many circumstances, the frequency of deleterious alleles in subdivided populations is expected to be lower than in an undivided population. If alleles were taken at random from a subdivided species and crossed, the load calculated would likely be much smaller than that in an undivided population. It is more biologically relevant, however, to calculate the load in the context of the breeding system of the species, accounting for the nonrandom mating associated with population structure. In principle, the load can be increased in a subdivided population even if
Similar derivations as above find the load for a haploid population to be
Note that, as in most discussions of load, this definition of load does not predict the decline in the mean number of offspring per individual actually observed in a population, because under soft selection the mean number of offspring is assumed to be constant per deme, and even under hard selection the mean productivity is constant for the species. These calculations would give the mean deficit of the relative fecundity of individuals from this species in competition with an individual without deleterious alleles or a hypothetical sister species.
Also note that these calculations are derived from the equilibrium values in an infinitely large metapopulation. With finite metapopulations, there is a chance that deleterious alleles will fix in the population (and therefore contribute to drift load) or that deleterious alleles are lost. The mutation load due to segregating alleles in species with a relatively small number of individuals is likely to be different from the values given here.
INBREEDING DEPRESSION
For h < ½ and s < 0, inbred individuals are likely to be less fit than relatively outbred individuals. This is because inbred individuals are more likely to express deleterious alleles as homozygotes than are outbreds. If organisms randomly chosen from a population are inbred such that their relative inbreeding coefficient is f, then their total inbreeding coefficient will be F_{TOT} ≡ 1 − ((1 − F_{ST})(1 − f)). The mean fitness of these inbred individuals is
With these simple definitions and using the approximation for
When inbreeding depression is calculated using the mean fitness of experimentally inbred individuals relative to the mean fitness of individuals experimentally outcrossed randomly across the metapopulation, the mean fitness of inbred individuals is as above, and the mean fitness of these outbred individuals is
By this standard, inbreeding depression will be lower in structured populations only for small values of h. See Figure 3B. This is important though, because loci with small values of h are responsible for a disproportionate fraction of inbreeding depression. Inbreeding depression is expected to be trivial for values of h near ½, so most inbreeding depression due to recessive alleles is likely due to the subset of mutations that have small values of h (see Figure 4). It is exactly in this range in which population structure has the strongest effect (see Figure 3, A and B). If, as a starting guess, new mutations had a uniform distribution of h between 0 and ½, then the reduction in inbreeding depression due to population structure can be dramatic (Figure 3C).
OVERDOMINANCE
With overdominance, heterozygotes are the most fit genotypes; therefore overdominance is another potential cause of inbreeding depression. Similar equations to those above can be derived to predict the evolution of overdominant loci in structured populations. With the fitness of each of the three genotypes defined as 1 − s:1:1 − t, the mean fitness of a deme with allele frequencies p and q=1p is
Solving for an equilibrium and assuming small η, we can find the mean allele frequency
Calculating the segregation load,
The inbreeding depression expected due to a locus with overdominant selection in a structured population can be calculated as above assuming η, s, t ⪡ 1, giving for δ_{1} and δ_{2}, respectively,
FINDING THE F_{ST} OF SELECTED LOCI
The preceding calculations are useful only if we know the value of F_{ST} for selected loci. In general this is a difficult task, but for weak purifying selection the F_{ST} of loci that are under uniform selection may be closely approximated by the F_{ST} of a neutral locus under the same population structure. This can be quantified under Wright's island model, using Wright's distribution of allele frequencies among populations (Wright 1937a,b). Assuming mutation to be weak relative to migration, we can find E[q^{2}] as
Similar calculations with overdominance show that if Ns, Nt ⪡ 1, then F_{ST} of selected loci will be closely approximated by a neutral F_{ST} if s, t < m. Here the assumption of uniform selection has been quite important. With balancing selection, locally variable selection, or frequencydependent selection, or even relatively weak selection may potentially cause F_{ST} to deviate from its neutral expectation.
LOCAL INBREEDING
So far, we have assumed that each local population mates at random. When this restriction is lifted, the alleles within an individual can be correlated relative to other alleles in the deme, which is reflected in Wright's local inbreeding coefficient, F_{IS}. In this section results are derived that allow for this local inbreeding.
With local inbreeding, each local population is not in HardyWeinberg proportions. The mean local fitness is then
These equations match those from Caballero et al. (1991) and Ohta and Cockerham (1974) for the case of local inbreeding within an undivided population. With hard selection, the only effect of population structure is a contribution of F_{ST} to the total inbreeding coefficient term. With soft selection, however, F_{ST} also affects the change in allele frequency by selection by affecting local competition, which is unlike the effects of F_{IS}.
For deleterious mutations (s < 0) at an equilibrium between mutation and selection,
DISCUSSION
Efficiency of selection: Many if not most species are to some extent subdivided into local populations, in which individuals are more likely to breed and/or compete with nearby individuals, who are more likely to be related to each other than are randomly chosen members of the species. These simple facts change the way in which even the simplest selection acts to affect allele frequencies, mutation load, and the inbreeding depression that might result. With locally biased mating, the additive genetic variance within populations tends to be reduced for additive alleles but can be increased with rare recessive or overdominant alleles (Robertson 1952; Tachida and Cockerham 1987, 1989; Whitlocket al. 1993; Willis and Orr 1993). Selection within populations therefore tends to be less effective in changing the frequencies of additively interacting alleles but can be more effective for rare recessive alleles. The genetic variance among populations and the total genetic variance in the species tend to increase with population structure, for the same overall allele frequency. Therefore, even with additively acting alleles, if populations are allowed to vary in their contribution to the next generation (as it would be in the case of hard selection), then a structured population will have more efficient response to selection than a panmictic one. The balance between these two effects—the change (up or down) in the response to selection within populations and the increase in efficiency of selection among populations—gives the overall effect of population structure on the change in allele frequency due to selection. Whether Δq is greater or less than expected in a panmictic population depends on whether individuals from different demes compete for resources (hard vs. soft selection), what the dominance relationships are between alleles, and the extent of genetic differentiation among populations.
For recessive alleles, the difference in response to selection can be substantial, even for relatively weak population structure. This difference is due largely to a change in the typical pattern of expression of the recessive alleles. With local mating, rare alleles are more likely to be expressed as homozygotes, and therefore the response to selection on recessive alleles will be in proportion to their homozygous effects rather than their weaker heterozygous effects. For hard selection, the change in the effects of selection and its consequences to load and inbreeding depression turn out to be exactly as would be expected from treating the nonrandom mating as a form of inbreeding (as in, for example, Workman and Jain 1966; Crow and Kimura 1970; Ohta and Cockerham 1974; Lande and Schemske 1985). With any soft selection, the resulting competition among relatives causes population structure to have unique effects.
Genetic load: Since Haldane (1937) and Muller (1950) proposed that the mean fitness of a population might be substantially reduced by “our load of mutations,” a great deal of argument has tried to resolve whether the rate of mutation to deleterious alleles is sufficient to cause the mean fitness of populations to be dangerously low (Crow and Kimura 1964; Crow 1993; Lynchet al. 1999).
If the genomic mutation rate to deleterious alleles is represented by U, then the mutation load due to partially dominant, multiplicatively interacting deleterious mutations in a large panmictic population is expected to be 1 − e^{−}^{U} (Crow 1993). Thus if the genomic mutation rate approaches unity or higher (see EyreWalker and Keightley 1999; Lynchet al. 1999), the mutation load could be quite large (e.g., 63% for U = 1; 93% for U = 2.7). This has led to the exploration of various deviations from these basic assumptions, as this load is thought to be too large to be borne by many species. In particular, a great deal of attention has been paid to the idea that deleterious mutations might interact synergistically, so that the rate of loss of fitness increases as the number of mutations goes up. While it is true that synergistic epistasis can significantly reduce mutation load in theory (Kimura and Maruyama 1966; Kondrashov and Crow 1988), there is little empirical support for the hypothesis that deleterious mutations interact in this way consistently (Whitlock and Bourguet 2000 and references therein). Others suggest that load may be reduced by intraindividual selection (Otto and Orive 1995) or sexual selection (Whitlock 2000). It may be that reproductive excess in resourcelimited populations is sufficient to allow for substantial load without extinction (Wallace 1991). It is also possible that for most organisms the genomic deleterious mutation rate is not so large (GarcíaDoradoet al. 1999). Crow and Kimura (1970) showed that the expected load at equilibrium with inbreeding can be reduced. Added to these explanations now is the hypothesis that a substantial amount of the possible genetic load may be eliminated by population structure. New analyses and experiments show that new mutations tend to be recessive with a mean dominance coefficient in the area of 0.1–0.2 (Hughes 1995; Houleet al. 1997; GarcíaDorado and Caballero 2000). In this range, the mutation load can be markedly reduced by even mild population structure, especially under hard selection.
A striking difference caused by population structure in the results for load is that the mutation load contributed by a locus is no longer independent of the genetic details of that locus. In particular, the dominance coefficient is now an important determinant of the mutation load, unlike the random mating case. Recessive alleles are likely to contribute less than codominant ones to the total mutation load.
The reduction in the overall frequency of deleterious recessive alleles and the expected decrease in the mutation load experienced by a subdivided population have much in common with the phenomenon of purging in bottlenecked or inbred populations. In bottlenecked or inbred populations, there can be a temporary reduction in the mutation load (Barrett and Charlesworth 1991; Byers and Waller 1999; Wanget al. 1999; Bataillon and Kirkpatrick 2000; Kirkpatrick and Jarne 2000), although experimental results are mixed (Byers and Waller 1999; Fowler and Whitlock 1999). These reductions in load are temporary, however, as mutation continues to increase load until it is returned to prebottleneck levels (Kirkpatrick and Jarne 2000). In structured populations, however, purging is possible because of the increased expression of homozygous individuals, but the effect is not temporary because the populations continue to be somewhat interconnected. New variation is brought into each population by migration, so the purging does not stop as it does in inbred lines. [This effect is similar to the pattern observed by Wang (2000) with alternate outcrossing and fullsib mating.] The deleterious allele frequency is therefore allowed to reach a different equilibrium, with fewer deleterious alleles and potentially a lower mutation load than in an undivided population.
The maximum reduction in load is by a factor of onehalf (with hard selection and nearly completely recessive alleles). This reflects the fact that in this case, most individuals that die a selective death are homozygotes, taking two deleterious alleles from the population for each selective death. Genetic load is a simple function of the number of individuals dying selective deaths and the number of deleterious alleles that die with them (Kondrashov and Crow 1988). If the number of alleles removed by each selective death could be doubled, then the genetic load is halved, as is almost the case in this example.
With overdominance and segregation load, however, the situation is reversed. Here the extra homozygosity caused by population structure results in a greater deviation from the maximum mean fitness, with an increase in load in proportion to F_{ST} (see Equation 33). This change in segregation load is surprisingly independent of whether the population experiences soft or hard selection. The relationship between load in subdivided and undivided populations is therefore likely to depend on whether mutation load or segregation load is more important.
It is important to note that the changes in genetic load that accompany population structure are not immediate. A previously undivided species that is suddenly subdivided will not change immediately to have a lower frequency of deleterious alleles, but it will quickly come to have a higher homozygosity. As a result, a newly formed metapopulation will be expected to have some inbreeding depression and a lower fitness than either an undivided species or a metapopulation at equilibrium. Therefore, for example, the fragmentation that results from human impact on the landscape is likely to have deleterious effects in the short to medium term.
Inbreeding depression: Inbreeding depression is likely due to a combination of the expression as homozygotes of rare recessive alleles maintained by mutationselection balance and a reduction in the number of overdominant heterozygous loci. These two patterns are called the dominance and overdominance models, respectively. With the dominance model, inbreeding not only reduces fitness but also allows the population to purge deleterious alleles to some extent, such that subsequent inbreeding may not display as much inbreeding depression (Lande and Schemske 1985; Byers and Waller 1999; Bataillon and Kirkpatrick 2000; Wang 2000). Overdominance depends on the presence of two (or more) alleles in the population, and the equilibrium allele frequency maximizes fitness in a panmictic population. Therefore purging is not possible with overdominance, and inbreeding can only reduce fitness, both immediately and ultimately. Therefore experimental metapopulations with hard selection will be a useful way of discriminating between the dominance and overdominance models of inbreeding depression. An experiment that artificially created metapopulations with hard selection from a previously undivided species should, at equilibrium, show much reduced inbreeding depression if the dominance model is prevalent, but only slightly reduced inbreeding depression if the overdominance model is most important. (The caveat to this is that if much inbreeding depression is due to overdominant loci with very asymmetric homozygous effects, with population structure these loci can fix for the allele corresponding to the fitter homozygote and the associated inbreeding depression goes to zero.)
Population structure causes some inbreeding, due to the greater probability that individuals will mate with related individuals in the same deme. This may reduce inbreeding depression in two ways: first, because the fitness of standard “outbred” individuals may reasonably be measured relative to typical individuals in the species, which are themselves somewhat inbred; and second, by changing the allele frequencies of deleterious alleles in the species as a whole. The two measures of inbreeding depression discussed in this article include both of these effects (in the case of δ_{1}) or just the second (in the case of δ_{2}). The extent of change in inbreeding depression in metapopulations depends on exactly how it is measured.
The inbreeding associated with population structure allows purging of rare, recessive, deleterious alleles and therefore can reduce the inbreeding depression due to dominance but cannot much affect the inbreeding depression due to overdominance. With overdominance, there is a smaller reduction in inbreeding depression due solely to the fact that the mean fitness of outbred individuals is somewhat reduced, because of the deviation in subpopulations from the allele frequencies that give maximum mean fitness. The inbreeding depression in structured populations is expected to be lower than that in an undivided population, and it is much reduced for the inbreeding depression caused by rare, deleterious recessive alleles.
The extent of this reduction depends on the distribution of dominance coefficients among new mutations. Nearly recessive mutations are much affected by population structure, with great reductions in expected inbreeding depression expected even with relatively small values of F_{ST}. Mutations with higher values of h are less likely to be affected. There is very little information about some aspects of this important distribution. A simple theoretical observation may help. Alleles with dominance coefficients near ½ do not contribute much to inbreeding depression, even in a panmictic population, for the simple reason that the value of their heterozygotes is not much different from the mean of the homozygotes. As h approaches 0, inbreeding depression is much greater (see Figure 4). As a result, even if the spectrum of new mutations includes few that are nearly recessive, it is these that disproportionately cause inbreeding depression, and therefore the largest effect of population structure occurs for the dominance coefficients that are most important.
One difficulty remains, however. A substantial fraction of inbreeding depression is caused by alleles of very large effect (Charlesworth and Charlesworth 1987). Such strong selection falls outside the conditions assumed in this article. While the inbreeding depression due to these large mutations, which tend to be nearly recessive (
Conclusions: Spatial population structure has often been studied, both theoretically and empirically, reflecting to some extent its prevalence in natural systems. We have measurements of F_{ST} from a wide variety of species. Recently, the argument has been made that F_{ST} is not a good measure of the rate of dispersal, the reason for which F_{ST} is often studied (Whitlock and McCauley 1999). This article, however, has shown the value of studying F_{ST} of even neutral loci and its power to predict interesting evolutionary processes. F_{ST}, as defined in this article, is an excellent description of the effects of spatial population structure on the response to weak selection.
The F_{ST} found to be most useful for these results differs, however, from its standard definition. The parameter needed gives equal weight to all individuals and does not necessarily weight populations equally. In this respect, it differs from other definitions in the literature. To simplify the mathematics, many theoretical models of F_{ST} assume equal population sizes at the point of measurement [e.g., the island model (Wright 1931), the stepping stone models (Kimura and Weiss 1964), and the basic extinctionrecolonization models (Whitlock and McCauley 1990)]. These models will predict the F_{ST} required here, but more work is needed to define this weighting for other population structures. More importantly, current estimates of F_{ST} from data weight populations equally, independent of their size. For many systems with variance in population size, empirical estimates of F_{ST} do not exactly match what is needed. Statistical work allowing the estimation of this weighted F_{ST} will therefore be very useful and straightforward.
Population structure allows an increase in homozygosity and competition among relatives, which both can change the dynamics of selection. As a result, the effects of selection can be weakened by local competition, but, perhaps more importantly on balance, selection can be intensified by the increase in genetic variance associated with greater homozygosity. As a result, equilibrium mutation load, inbreeding depression, and the rate of response to selection can be changed, often in a way beneficial to the fitness of the population.
Acknowledgments
I thank S. P. Otto, F. Rousset, M. Wade, S. Glémin, and two anonymous reviewers for very helpful comments on the manuscript and N. Barton, D. Charlesworth, J. Ronfort, and Y. Michalakis for discussions. This work was funded by the Natural Science and Engineering Research Council (Canada) and was done in part while I was kindly hosted at the Institute for Cell, Animal and Population Biology at the University of Edinburgh and at the Institut de Recherche pour le Développment in Montpellier, France, funded by the Centre National de Recherches Scientifiques. Many thanks to them all.
APPENDIX: γ AND THE SKEWNESS OF THE ALLELE FREQUENCY DISTRIBUTION
Let us define μ_{3} to be the skewness of q among populations. If we define a term
We can calculate the value of γ in an island model. Wright (1937a,b) determined the distribution of allele frequencies across populations for the neutral island model without mutation,
Footnotes

Communicating editor: D. Charlesworth
 Received May 31, 2001.
 Accepted November 27, 2001.
 Copyright © 2002 by the Genetics Society of America