Abstract
Inbreeding depression is a general phenomenon that is due mainly to recessive deleterious mutations, the socalled mutation load. It has been much studied theoretically. However, until very recently, population structure has not been taken into account, even though it can be an important factor in the evolution of populations. Population subdivision modifies the dynamics of deleterious mutations because the outcome of selection depends on processes both within populations (selection and drift) and between populations (migration). Here, we present a general model that permits us to gain insight into patterns of inbreeding depression, heterosis, and the load in subdivided populations. We show that they can be interpreted with reference to singlepopulation theory, using an appropriate local effective population size that integrates the effects of drift, selection, and migration. We term this the “effective population size of selection” (
INBREEDING depression, the decline of fitness of inbred individuals relative to outbred ones, is a general phenomenon observed in many species (Charlesworth and Charlesworth 1987) and for a long time (Darwin 1876). It has been much studied theoretically (Lande and Schemske 1985; Charlesworthet al. 1990b; Bataillon and Kirkpatrick 2000) and experimentally (Schemske and Lande 1985; Husband and Schemske 1996) because it is supposed to play a key role in the evolution of mating systems and to challenge the viability of small populations. The genetic basis of inbreeding depression has been extensively investigated and it is now recognized that it is due mainly to deleterious and partially recessive mutations, even if polymorphism maintained by balancing selection may also play a role (Charlesworth and Charlesworth 1999). The mutation load is often defined as the decline of mean fitness due to mutation accumulation relative to an ideal population free of mutation (Crow 1970). In very large populations, the mutation load depends only on the genomic mutation rate (often referred to as U; Haldane 1937), while the magnitude of inbreeding depression depends on U and on the levels of dominance of the mutations (Charlesworth and Charlesworth 1987). Fully recessive mutations are maintained in higher frequencies than partially recessive ones and thus cause greater declines in fitness under consanguineous matings. Inbreeding depression can be easily estimated by comparing the performances of progenies produced by outcrosses vs. consanguineous crosses. On the contrary, one cannot estimate the load directly because the ideal reference population does not exist. A better knowledge of the load and inbreeding depression can be obtained by characterizing the properties of deleterious mutations (mutation rates, level of dominance, and deleterious effect) and different methods have been proposed to estimate them (for review, see Deng and Fu 1998; Bataillon 2000a). One method relies upon mutation accumulation experiments (Mukaiet al. 1972), whereas the others use measures of inbreeding depression or some equivalent (Charlesworthet al. 1990a; Deng and Lynch 1996; Deng 1998).
In methods using measures of inbreeding depression, the underlying models neglect two potentially important factors: population size and population structure. Nevertheless, population size and genetic drift may have a huge impact on the expected inbreeding depression due to deleterious mutations. Moreover, drift has opposite effects on the load and inbreeding depression: the load is higher in small populations than in large ones (Kimuraet al. 1963) while the reverse pattern is expected for inbreeding depression (Bataillon and Kirkpatrick 2000). Population subdivision introduces two additional complications for studying the effects of deleterious mutations. First, the outcome of selection is dependent on process both within populations (selection and drift) and between populations (migration). Second, crosses and their fitness can be defined at different scales and the choice of a reference is thus crucial, as already pointed out by Waller (1993) and Keller and Waller (2002).
Some experimental studies have attempted to estimate inbreeding depression and/or heterosis in subdivided populations. Heterosis can be defined as the excess in mean fitness of individuals produced by crosses between demes relative to mean fitness of individuals produced by outcrosses within deme. Some studies have addressed population levels and hierachical measures of inbreeding depression (Ouborg and Van Treuren 1994; Carr and Dudash 1995; Byers 1998; Richards 2000; Sheridan and Karowe 2000) while others have conducted only global analysis (Saccheriet al. 1998; Van Oosterhoutet al. 2000). In both types of studies, inbreeding depression and the mutation load are often not clearly distinguished. Indeed, until very recently, the lack of theoretical predictions on the expected patterns of inbreeding depression, heterosis, and the load in subdivided populations was patent. Selection in subdivided populations has already been investigated (Maruyama 1972a,b,c; Nagylaki 1989). However, to our knowledge, only two recent studies have focused on the patterns of inbreeding depression in subdivided population. Theodorou and Couvet (2002) have used numerical computations to study specifically the joint effect of selfing and population subdivision on the evolution of inbreeding depression. Whitlock (2002) has developed an analytical method for large metapopulations and weak selection, adressing the outcome of selection and its prediction using neutral F_{ST}.
Here, we present a general method to study the pattern of inbreeding depression, heterosis, and mutation load expected for a broad range of population structure. We have adapted a twolocus diffusion method, developed by Ohta and Kimura (1969, 1971), to a onelocus treatment in multideme systems. We obtain analytical results in the case of strong selection that naturally lead to the definition of a new effective population size, which integrates the effects of selection, drift, and migration. The pattern of inbreeding depression, heterosis, and the load can be comprehensively interpreted with reference to singlepopulation theory, using this effective population size. Our results suggest a way to define and estimate inbreeding depression and the load in subdivided populations. We also discuss the implications of our results for the evolution of mating systems and conservation issues.
MODELS AND RESULTS
General presentation
We consider a single locus with two alleles in a metapopulation of K demes, each composed of N diploid individuals, connected by migration. Individuals successively experience mutation and reproduction in each local deme. After zygotic migration, selection occurs within each local deme, followed by density regulation. The contribution of each deme to the next generation is constant and independent of the mean fitness of the deme. The wildtype allele, A, mutates at rate μ to a partially recessive, deleterious allele, a. The reverse mutation occurs at rate ν with ν ≪ μ. The relative fitnesses of the AA, Aa, and aa genotypes are 1, 1 – hs, and 1 – s, respectively, where s is the selection coefficient and h the dominance coefficient. For simplicity, we analyze only the case where h and s are identical across all demes. We first consider random mating in each deme. For a deleterious mutation segregating at frequency x_{i} in the ith deme, we can define the mean fitness of individuals produced by the different types of crosses. The mean fitness among individuals produced by outcrossing, in the ith deme,
In deme i, we define inbreeding depression, δ_{i}, as the decline in mean fitness of selfed individuals relative to outcrossed individuals within the deme (Charlesworth and Charlesworth 1987), and the genetic load, L_{i}, as the decline in the mean fitness of the deme relative to the optimal genotype (AA; Crow and Kimura 1970):
To obtain expected values for our load and inbreeding depression parameters (L, δ, H, γ), we have to compute the expectation of the four quantities previously defined over Φ(x_{1},..., x_{K}), the probability distribution of the deleterious allele frequency over the K demes of the metapopulation. One can use Wright's distribution (see Wright 1969) or the extensions given by Maruyama (1972b) for steppingstone models. However, because these distributions are implicitly defined, only numerical results can be obtained. Whitlock et al. (2000) followed this approach to investigate the magnitude of heterosis and drift load for the infinite island model. Here, we develop analytical approximations for the patterns of inbreeding depression in subdivided populations. If we are able to satisfactorily approximate δ, L, H, and γ by polynomial functions of degree p, their expectations over Φ will depend only on the p first moments of Φ. Practically, the two first moments are sufficient: the load is a quadratic function of x_{i} and good approximations of δ_{i}, H_{ij}, and γ_{ij} are obtained, assuming that
Analytical results for the case of strong selection (Nhs ≫ 1)
OhtaKimura equation for subdivided populations: To compute the first two moments of Φ, we adapted the method developed by Ohta and Kimura (1969, 1971) to study the linkage disequilibrium in twolocus models under mutationdrift equilibrium (see also Appendix 3 of Kimura and Ohta 1971 for details). This method has been used for twolocus problems in different situations (e.g., see Petry 1983; Nordborget al. 1996), but, to our knowledge, this is the first time that it has been adapted to model subdivided populations.
According to diffusion theory, for each deme, we need to write the following infinitesimal terms: the mean change of allele frequency, M_{δ}_{xi}, the variance of the change of allele frequency, V_{δ}_{xi}, and the covariance of the change of allele frequency in a pair of demes, W_{δxi,δxj}. M_{δxi} reflects mutation, migration, and selection:
Assumptions for solving the system: Δ_{mut}(x_{i}) and Δ_{mig}(x_{i}) are linear terms but not Δ_{sel}(x_{i}). To satisfy the linearity condition on M_{δ}_{xi}, we linearized the selection term in 0 (x_{i} ≪ 1) following Robertson (1970) and Bataillon and Kirkpatrick (2000): Δ_{sel}(x_{i}) =–hsx_{i}. This is equivalent to assuming that selection acts only against heterozygotes. This assumes that deleterious alleles are not too recessive (h > 0) and maintained in low frequencies; so it is also assumed that μ ≪ hs. This approximation is thus valuable only if local drift is not too strong. In a single population these conditions correspond about to Nhs > 5 (Bataillon and Kirkpatrick 2000). With low migration rates, the analyical results are thus valid for population sizes of the order of 100 at least. If migration overwhelms local drift (high migration rates), we might expect that x_{i} will be in low frequency in all demes such that the Nhs limit can be lower. Accuracy of the approximation is now tested further against numerical or simulation results.
The Kisland model: Computation of the moments: We consider K panmictic demes of size N, connected by migration at a rate m. The infinitesimal diffusion terms are given by Equations 4a, 4b, 4c with (4a) becoming
For the function f(x_{1},..., x_{K}) = x_{i}, Equation 5 implies, for the stationary distribution,
For the function
We can also compute the moments of the distribution of deleterious allele frequencies over the whole metapopulation, Ψ. The frequency, y, of the deleterious allele in the whole metapopulation is
“Effective population size of selection”: Using the same approximation [linearization of Δ_{sel}(x_{i})] Bataillon and Kirkpatrick (2000) have shown that, in a single large but finite population, the deleterious allele frequency follows a β distribution with mean x̂ =μ/hs + O(μ^{2}) and variance σ^{2} =μ/hs(1 + 4Nhs) + O(μ^{2}). Comparing this result to Equations 8 and 11a, we note that each local deme under the infinite island model is equivalent to a single population under this model, with a new local effective population size. We chose to call this parameter “effective population size of selection.” Extracting N from the expression for σ^{2}, we can generally define this parameter:
For the Kisland model,
Using (8), (10a), and (10b), we find for the Kisland model
Average inbreeding depression, genetic load, and heterosis: Using Equations 3a, 3b, 3c, 3d, the expressions for the first and secondorder moments and the expressions for the
Inbreeding depression is given by
In the same way, we can compute the average load,
Heterosis is given by
These derivations show that population subdivision has opposite effects on inbreeding depression within and between demes. It decreases local inbreeding depression, compared to an infinite population, whereas it increases betweendeme inbreeding depression. As expected, heterosis also increases with subdivision. According to the expression of
Results for the load are slightly inaccurate because we have neglected the mild purging effect that occurs in finite but not too small populations under weak subdivision when h < ⅓ (see S. Glémin, unpublished results; and Whitlock 2002). This purging effect has only weak quantitative consequences on Equation 19 but some qualitative consequences under h < ⅓. For h > ⅓, Equation 19 is quite accurate (see Figures 4 and 5). For h < ⅓, the load decreases with weak subdivision before increasing when subdivision is more important (see also Whitlock 2002). However, these variations are weak relative to the strong increase of the load in small populations (Kimuraet al. 1963).
The infinite island model with nonrandom mating: With nonrandom mating, we need more general expressions for inbreeding depression, the genetic load, and heterosis as a function of the moments of the probability distribution of deleterious allele frequency. The various mean fitnesses of interest can be expressed as functions of the deleterious allele frequency and fixation index F_{IS} (Caballero and Hill 1992). We assume that all demes have the same F_{IS}. Within the ith deme,
Computation of the moments: Following Caballero and Hill (1992) and Bataillon and Kirkpatrick (2000) with the addition of migration, the infinitesimal diffusion terms are obtained from Equations 6, 4b, and 4c by changing h to h_{F} = (h + F_{IS} – hF_{IS}) and N to N_{e} = N/(1 + F_{IS}). Here, the linearization of the selection term is equivalent to assuming that selection acts only on heterozygotes produced by random mating and against homozygotes produced by nonrandom mating.
Using Equation 5 for the same appropriate f functions and considering the symmetrical properties of the island model, we can compute the first and secondorder moments of Φ in the case of nonrandom mating. All moments are found to be the same as in the panmictic case (see Equations 8, 10a, and 10b), after replacement of h by h_{F} and N by N_{e} = N/(1 + F_{IS}).
Effective population size of selection and selected
Average inbreeding depression, genetic load, and heterosis: Using Equations 23a, 23b, 23c, 23d and the expressions for the first and secondorder moments, we can now compute the average local inbreeding depression and genetic load and the average heterosis and inbreeding depression between two demes. For inbreeding depression, Equation 13 is still valid using the appropriate
The expression for the load is different from (18),
Inbreeding depression between demes and heterosis are now given by
Inbreeding due to nonrandom mating (F_{IS}) decreases both inbreeding depression and the load (as in an infinite population). It also decreases heterosis and inbreeding depression between demes (see Figure 6). With high levels of inbreeding, the effect of migration on the load and inbreeding depression (within and between demes) is very weak (see Figure 6A for inbreeding depression). The effect of migration on heterosis is more important, even with inbreeding (see Figure 6B). For weak selection, equivalent results have been obtained independently with numerical methods by Theodorou and Couvet (2002) for inbreeding depression and the load. However, they found that for high inbreeding levels, migration has also very little effect on heterosis.
The unidimensional steppingstone model: We now consider a circular steppingstone model with K panmictic demes of size N, with K = 2p or K = 2p + 1. We assume only local and equal migration between two adjacent demes. We now need to compute p + 2 moments: E_{Φ}[x], E_{Φ}[x^{2}], as in the previous cases and the secondorder interdeme moments, E_{Φ}[x_{i}x_{i}_{+}_{k}] for a pair of demes separated by k steps, which depend on the distance between demes. All the moments between two demes at distance k are equal and we denote them E_{Φ}[xx_{k}]. We (see appendix b). Here, we give the results for the infinite unidimensional steppingstone model (K → ∞):
We can also compute
Robustness and generalization of the analytical results: The results above are not valid for weak selection. Indeed, because of genetic drift, the frequency of a deleterious allele can be high (near 1), so we cannot linearize Δ_{sel}(x_{i}) around x_{i} = 0; i.e., selection against homozygotes aa cannot be neglected. However, we can extend qualitatively our theory to more general sets of parameters. The weakness of our approximations is that drift and subdivision do not affect the mean frequency of the deleterious allele, which is always μ/h_{F}s. However, the variance of the frequency of the deleterious allele, which leads to the definition of the effective size of selection, is much better predicted by the theory (see Table 1). Consequently, as we have already said, the expression for
Qualitative patterns of the load due to segregating mutations are well predicted by
DISCUSSION
Heuristic value of the theory and limitations of the model: In this study, we adapted a diffusion method to provide general and analytical results to understand the effects of population subdivision on patterns of inbreeding depression, heterosis, and the load due to partially recessive deleterious mutations. Because this method leads to linear equations with respect to the moments of the distribution of allele frequency, any kind of subdivision can be studied. The more general and heuristic result we obtained is that one can use an index of effective size of selection to interpret the effect of subdivision by reference to singlepopulation theory. Accurate analytical results are obtained for strong selection. Our effective size of selection is still a useful index for a wider range of situations in which our analytical results may be less accurate. In a single population, deleterious alleles for which Ns ≫ 1 segregate in low frequency while those for which Ns < 1 can be nearly fixed. In a local deme of a subdivided population, qualitative predictions can be easily made using the same dichotomy but replacing N by
As in other models (Whitlocket al. 2000; Theodorou and Couvet 2002; Whitlock 2002), our predictions hold for one locus. Extrapolation for total fitness, which is of interest for the evolution of mating systems and conservation issues, must assume independence among loci. Under drift and subdivision this can be misleading because linkage disequilibrium can lessen the efficiency of selection (Hill and Robertson 1966). Under the assumptions we used (strong selection, Nhs ≫ 1), we expected that such associations should develop only between tightly linked loci. However, if there is a wide distribution of deleterious effects of alleles, interferences between weakly and strongly selected loci are more likely to happen (Stephanet al. 1999). Multilocus extension of such models is thus needed to answer to these questions.
The genetic basis of inbreeding depression and heterosis in subdivided populations: Inbreeding depression and heterosis are often seen as two aspects of the same genetic process. However, we show here that their genetic basis can be quite different. Inbreeding depression is primarily due to mutations with strong effect (for which
Contrary to this interpretation, one can argue that it depends on the definition of heterosis, which differs from that generally given by plant breeders: “When inbred lines are crossed, the progeny show an increase of those characters that previously suffered a reduction from inbreeding.... The amount of heterosis is the difference between the crossbred and the inbred means” (Falconer 1989, p. 254). Indeed, for completely autogamous populations, according to this definition, heterosis is the exact opposite of inbreeding depression. However, in all other cases, inbred lines have to be produced before outcrossing them. During recurrent selfing, purging of lethals and fixation of mildly deleterious alleles can occur. Consequently heterosis is not just the reverse of inbreeding depression that can be estimated in the base population. Each line can be viewed as a single isolated population and heterosis is well defined by the excess of mean fitness of individuals produced by outcrosses between demes (equaling lines) relative to mean fitness of individuals produced by outcrosses within demes (equaling mean fitness of a line).
Consequences for the evolution of mating systems in subdivided populations: Inbreeding depression is a key parameter in the evolution of mating system because it balances the “cost of outcrossing” (Fisher 1941; Uyenoyama 1986). The majority of theoretical studies modeling the evolution of mating systems assumes single and large (infinite) populations (but see Holsinger 1986). Here, we show that drift and subdivision can modify the patterns of inbreeding depression and heterosis and should influence the direction of selection on the mating systems. In our analysis, the mating system does not evolve but it determines, jointly with the pattern of migration, the amount of inbreeding depression. In subdivided populations, two opposite pressures can influence the evolution of selfing. Increased subdivision, just as drift, leads to a decline of withindeme inbreeding depression, which reduces the advantage of outcrossing. However, subdivision also reduces the cost of outcrossing (Uyenoyama 1986). Conversely, drift and subdivision also increase betweendeme inbreeding depression, which may favor outcrossing with migrants. For weaker selection against deleterious alleles (Ns < 1), Theodorou and Couvet (2002) found similar patterns of inbreeding depression and heterosis. Taking account of both of these patterns and the cost of outcrossing, they suggest that mixedmating systems should be stable for intermediate migration rates, especially with pollen migration (that is not assumed in our study). In this case, heterosis increases as the selfing rate increases (because selfing limits pollen migration). Advantage to migrants increases with selfing and can prevent the full invasion of a modifier causing selfing. With a stochastic model of selection on selfing rates in a continuous structured population, Ronfort and Couvet (1995) also show that population structure should maintain intermediate selfing rates.
However, as already noted, our model and others (Whitlocket al. 2000; Theodorou and Couvet 2002; Whitlock 2002) are basically singlelocus models and associations between loci are not taken into account. To understand the real impact of population subdivision on the evolution of mating systems, such associations in subdivided populations should be modeled. Moreover, migration rates can also evolve in metapopulations in response to sib competition and to spatiotemporal variability in population sizes (e.g., Olivieri and Gouyon 1997). Heterosis could also be another factor that selects for increasing migration rates (Morgan 2002). However, the coevolution of mating system, inbreeding depression, and migration is still a challenging question.
Implications for conservation biology: The accumulation of deleterious mutations in small populations should increase the risk of extinction due to the process called “mutational meltdown” (Lynchet al. 1995). A recent simulation study (Higgins and Lynch 2001) has shown that this process can also occur in metapopulations. The increase in the risk of extinction is mainly due to the drift load. A simple conclusion of our analysis is that migration between populations efficiently purges the main part of the load, by converting the drift load into load due to segregating mutations; i.e., more mutations can be selected against (see Figure 8). Moreover, longdistance migration purges the drift load more efficiently than local migration does (Higgins and Lynch 2001); the effective size of selection is higher in the island model than in the steppingstone one (see Figure 1). In our model, we neglect the fact that intermediate migration rates may favor the purging of recessive segregating alleles (see discussion above and Whitlock 2002). However, intermediate migration rates can reduce the load due to segregating mutations but increase the drift load (Whitlocket al. 2000). We thus think that for conservation of endangered populations, high connection among populations should be less risky and globally much better than maintaining intermediate migration rates. In addition, such optimal migration rates should be very difficult to estimate. Such a positive demographic effect of gene flow between populations, known as the “genetic rescue effect,” has been documented in metapopulations of Silene (Richards 2000) and Daphnia (Ebertet al. 2002; Haaget al. 2002). Advantages to migrants can increase the effective migration rates (see also Ingvarsson and Whitlock 2000) and protect some demes from extinction, especially small and isolated ones (Richards 2000).
Other implications of our analysis include methodological considerations. The load would be an appropriate measure for estimating the impact of population size and subdivision on the fitness of small populations. However, this quantity cannot be directly estimated. Experimental designs for the estimation of inbreeding depression have been proposed to address this question (Charlesworth et al. 1990a; Deng and Lynch 1996; Deng 1998). Bataillon and Kirkpatrick (2000) and others have already stressed that inbreeding depression is not a useful indication of the load in small populations. In subdivided populations this conclusion still holds. But our analysis shows that in subdivided populations, patterns of heterosis are similar to patterns of the load and may constitute a measure of the load more appropriate than inbreeding depression for conservation purposes. Moreover, mutations that cause the highest load also cause the highest heterosis but cause no inbreeding depression (Bataillon 2000b). More precisely, heterosis can provide an indication of the local drift load (see also Whitlocket al. 2000), i.e., the excess of load due to fixation of deleterious mutations in small populations compared to the load due to segregating mutations maintained in large ones. We thus propose that joint measures of inbreeding depression and heterosis provide a general picture of the architecture of the load in subdivided populations. The load due to segregating mutations could be estimated through inbreeding depressionbased methods as already proposed (Charlesworthet al. 1990a; Deng and Lynch 1996; Deng 1998). However, the present study shows that corrections for population size and migration should be taken into account. Alternatively, the drift load could be estimated through the measurement of heterosis. A limitation of this method is that we need to assume no local adaptation. Consequently, an analysis of heterosis would probably underestimate the drift load.
Few data that compare inbreeding depression and heterosis among populations of different sizes or degrees of isolation exist. Often, mean performances of whole populations of different sizes are compared (Eldridgeet al. 1999; Casselet al. 2001). However, a positive correlation between population size (and thus global inbreeding level) and mean fitness is not evidence for inbreeding depression, as is sometimes claimed, but evidence of increasing load in small populations. A recent study on the plant Silene alba (Richards 2000) is in good agreement with our predictions and clearly illustrates the distinction among inbreeding depression, heterosis, and the load. Germination rates were measured for sib crosses, outcrosses within and between sites, and two kinds of sites, central vs. isolated populations, were contrasted. Inbreeding depression was lower in isolated populations than in central ones (∼0.2 vs. 0.5). On the contrary, heterosis was high in isolated populations (∼0.6) but weak and nonsignificant in central populations (∼0.06). Finally, germination rates for outcrosses within populations were higher in central than in isolated populations, which indicates a higher load in isolated populations. Estimation of the load through inbreeding depression would lead to the reverse conclusion, that central populations suffer higher load than isolated ones. The comparison of inbreeding depression and heterosis shows that the architecture of the load differs between central and isolated populations. The load is weak and mainly due to segregating mutations in central populations whereas it is quite high and mainly due to fixed mutations in isolated populations. We thus suggest clearly distinguishing among inbreeding depression, heterosis, and the load in experimental studies, through the determination of the levels of population structure at which fitnessrelated traits are compared. This should allow easier comparison among studies and more complete analysis of the consequences of deleterious mutations in natural populations.
Acknowledgments
We thank M. Whitlock, K. Theodorou, and D. Couvet for helpful discussions and comments and for providing us their manuscripts before publication. We also thank N. Bierne and M. Uyenoyama and two anonymous reviewers for comments on the manuscripts and F. Rousset and I. Olivieri for helpful discussions. S.G. acknowledges a Ph.D. grant from the French Ministry of Education and Research. This work was supported by the Bureau des Ressources Génétiques, by the French Ministère de l'Aménagement du Territoire et de l'Environnement through the national program Diversitas, fragmented population network (contract 98/153), as well as by the European Community Fragland project (headed by I. Hanski). This is publication ISEM 2003051 of the Institut des Sciences de l'Evolution de Montpellier.
APPENDIX A: F ST S AS A FUNCTION OF THE MOMENTS OF Φ
The total genetic variance of allele frequency over the whole metapopulation,
APPENDIX B: MOMENTS OF THE DISTRIBUTION OF DELETERIOUS ALLELE FREQUENCY IN THE STEPPINGSTONE MODEL
We consider a circular steppingstone model with K panmictic demes of size N, with K = 2p or K = 2p + 1. We assume local (and equal) migration between two adjacent demes. The diffusion terms are
Secondorder moments: Using the function
System solving: The system can be reduced to one general recurrent equation and two specific “boundary” conditions:
For i = 1,... p – 1,
To determine a and b we use Equation B7b for i = 1 and i = p – 1, replacing E_{Φ}[x^{2}] and E_{Φ}[xx_{p}] by their expressions given by (B7a) and (B7c). We thus obtain two linear equations in a and b:
APPENDIX C: NUMERICAL COMPUTATION OF WRIGHT'S EQUATION
For the infinite island model, we can compute numerical values of inbreeding depression, mutation load, and heterosis, using Wright's distribution for Φ,
Equation (C1) can be numerically solved by iteration of the integration of
To test the usefulness of the effective size of selection, we also use Wright's equation for a single population, replacing N_{e} by
In both cases, direct calculations were done using the function NIntegrate of the mathematical software package Mathematica (Wolfram 1996).
Footnotes

Communicating editor: M. Uyenoyama
 Received June 11, 2002.
 Accepted August 10, 2003.
 Copyright © 2003 by the Genetics Society of America