NONSUBDIVIDED POPULATION

We assume that two alleles A and a at a given locus segregate in a population of N monoecious individuals. We call p the frequency of A, and q = 1 − p the frequency of a. At the beginning of each generation, each of the N adults produces a very large number of gametes and dies; we assume that allele A has a deleterious effect, such that aa, Aa, and AA individuals produce a number of gametes proportional to 1, 1 − hs, and 1 − s, respectively. Self-fertilization occurs at a rate α; more precisely, a proportion α of fertilizations involve two gametes produced by the same individual, while the other 1 − α involve two gametes taken at random among all gametes produced (therefore when α = 0 self-fertilization occurs at rate 1/N). Finally, N individuals are sampled randomly among the large number of juveniles produced to form the next adult generation. At each generation, mutation from a to A occurs at rate u, while mutation from A to a occurs at rate v.

The mutation load L is defined as the reduction in mean fitness of a population due to the constant input of deleterious mutations (Crow 1970). Calling p_aa, p_Aa, and p_AA the frequencies of the three genotypes in the population, we have 1

Genotypic frequencies are expressed as functions of p, the frequency of allele A, by the relations p_aa = q² + Fpq, p_Aa = 2(1 − F)pq, and p_AA = p² + Fpq, where F is the correlation between uniting gametes due to nonrandom mating (Wright's F_IS). Although F depends on the selection coefficient, it is sufficient to evaluate it in the neutral case to obtain an expression of the load to the first order in s; under neutrality and with a rate of selfing α, F = α/(2 − α) (e.g., Gillespie 1998, p. 93). This gives an expression of the load as a function of p: 2

Inbreeding depression, δ, is defined as the reduction in fitness of an individual produced by selfing, relative to the fitness of an individual produced by outcrossing (Charlesworth and Charlesworth 1987), which gives 3giving 4

In a population of finite size, at equilibrium, allele A will be in frequency p with a given probability φ(p). To obtain the mean load and inbreeding depression at equilibrium, one has to replace p and pq in Equations 2 and 4 by ∫pφ(p)dp and ∫pqφ(p)dp. An approximation for φ(p) can be obtained from diffusion methods: assuming that N is large and that s, u, and v are small (of order 1/N), the change in frequency of A over generations can be approximated by a diffusion process, with drift and diffusion coefficients 5 6

(Caballero and Hill 1992) with 7

M_δp and V_δp measure the mean and variance of the change in frequency of A over one generation; φ(p) is expressed as a function of these two quantities by the relation 8

(e.g., Ewens 1979), where K is a constant such that probabilities sum to one. This gives 9

This distribution has to be integrated numerically to obtain its first two moments and the average values of the mutation load and inbreeding depression. This was done by Bataillon and Kirkpatrick (2000), who showed in particular that the mean load is a decreasing function of population size, while the mean inbreeding depression increases with population size, these effects being marked mostly when population size is small. Some insights can be gained by considering the values of p, L, and δ in the limit as population size tends to infinity (deterministic equilibrium); these values can be obtained simply by solving M_δp = 0 for p and injecting this equilibrium value of p in Equations 2 and 4. Simpler expressions are obtained by neglecting back mutation (v = 0); one may also neglect terms in p², at least when h > 0, while when h = 0 these terms have to be conserved. This leads to different solutions for the cases h = 0 and h ≫ 0; these solutions are given in Table 1 for α = 0 (no self-fertilization), and α > 0. From the expressions given in Table 1, one finds easily that self-fertilization decreases the frequency of deleterious mutants and the mutation load as long as 0 < h < 1 and decreases inbreeding depression as long as 0 ≤ h < ¹/₂. One can also see that while the load depends only on u under random mating, it also depends on h and α when self-fertilization occurs (Ohta and Cockerham 1974).

POPULATION SUBDIVISION

We now turn to the case of a subdivided population. We assume an island model of population structure, where n is the number of demes and N the number of adult individuals per deme. Adult individuals produce a large number of gametes, still in relative proportions 1, 1 − hs, and 1 − s depending on their genotype. Fertilization occurs within each deme with a rate α of selfing, and the juveniles produced migrate at a rate m. We distinguish soft and hard selection as follows: under soft selection, the number of juveniles per deme is regulated to a constant value (the same for all demes) just before migration, while under hard selection there is no such regulation, and thus the number of juveniles produced by a deme at the time of migration depends on the genotypes of the parents that were in that deme (Christiansen 1975, Equation 1; Nagylaki 1992, p. 134); other definitions of soft and hard selection exist in the literature (Wallace 1975; Whitlock 2002). Finally, once migration has occurred, N individuals are sampled in each deme to form the next adult generation.

RESULTS

Approximate solutions

Although we could not find any exact solution for the mean of p and pq over a probability distribution of the form of Equation 9, some approximations can be obtained, as explained in appendix B. A first approximation, valid when h > ∼0.3, or when h < 0.3 and m is small (see discussion at the end of appendix B for more details), is 33

This approximation also corresponds to the deterministic solution (very large number of demes), obtained after neglecting terms in p². When h < 0.3, a better approximation is 34

(from appendix B), where . In (33) and (34), N_e is given by Equation 28, while S₁ and S₂ are given by (25) and (26) if selection is soft and by (29) and (30) if selection is hard. Another aproximation for the case h = 0 and m not too small is given in appendix B, Equation B11.

Figure 1 compares these approximations with numerical integrations using the NIntegrate function of Mathematica 4.1 (Wolfram 1991). When h = 0, (33) and (34) give good results only for small values of m; as h increases, the range over which they are accurate increases, (34) giving better results than (33), while when h > 0.3, (33) gives good results for all values of m. Figure 1 corresponds to the case of random mating within demes (α = 0). We observed that the accuracy of approximations (33) and (34) increases as the selfing rate α increases; approximation (B11), however, gives good results only for small values of α (results not shown). Finally, we observed that the expression of the mutation load obtained from equations (13) and (33) gives good results for all values of h, even when h = 0 and m is large (not shown).

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Figure 1.—

Average frequency of allele A (left) and within-deme inbreeding depression (right) in a subdivided population under hard selection, relative to the frequency of A and the inbreeding depression in a nonsubdivided population of the same total size, as a function of the migration rate (x-axis) and the dominance coefficient. Parameter values are n = 200, N = 100, s = 0.01, α = 0, u = 2 × 10⁻⁵, v = 10⁻⁶; top, h = 0; middle, h = 0.1; bottom, h = 0.3. p̅ and p̅q̅ were obtained by numerical integration over the φ distribution (thick solid lines), using approximation (33) (dashed lines), approximation (34) (dashed/dotted lines), or approximation (B11) (thin solid lines). The dotted lines correspond to approximations (35) and (37).

Approximations (33) and (34) are complicated functions of the parameters, as they depend on r₀, r₁, a, and c, which are the solutions of the equations given in appendix A. When m is small and N large, however, these solutions can be approximated by Equations 31 and 32, which leads to a simple expression for S₁. In that case, using (33) leads to (for both hard and soft selection) 35 36 37 38 39

Equations 35 and 36 can also be obtained from Whitlock's (2002) expressions for the mutant frequency and mutation load in an infinite-island model under soft selection (Equations 43, 44, and 46 in Whitlock 2002, with b = 0). Under hard selection we still obtain Equations 35–37 while Whitlock finds different results; this comes from the fact that our definition of hard selection is different from that in Whitlock (2002)(see Roze and Rousset 2003).

Quantitative patterns

Effects of population structure:

Figure 2 shows the average frequency of the deleterious allele A as a function of the migration rate m, under hard selection (results under soft selection are qualitatively very similar), when α = 0. Here we integrated numerically over the distribution given by Equation 9, with N_e, S₁, and S₂ given by Equations 28–30, and r₀, r₁, a, and c calculated from the recurrence equations given in appendix A. One can see from Figure 2A that the mutation-selection balance when m = 1 is not exactly the same as in a nonsubdivided population of the same total size; this comes from the fact that even when m = 1, population structure still has the effect of increasing the probability of self-fertilization, as we assume that fertilization occurs within demes before migration. This effect becomes negligible when deme size is large. For all parameter values that we tested, we found that reducing the migration rate up to a given point reduces the frequency of deleterious mutants when h < ∼¹/₃, and increases it when h > ∼¹/₃, which corresponds to what Whitlock (2002) had found in his soft-selection model. For small values of the migration rate, however, the frequency of deleterious mutants increases as m decreases, for all values of h, as can be seen on the left of Figure 2A. Indeed population structure has different effects on the selection against deleterious alleles: it increases homozygosity, which helps to purge recessive deleterious alleles, but it also decreases the efficiency of selection, by increasing the genetic similarity among competing individuals. When migration is very weak, demes are almost always fixed for a or A and selection has little effect; this explains why the curves go up on the left of Figure 2A.

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Figure 2.—

(A) Average frequency of allele A in a subdivided population under hard selection, relative to its average frequency in a nonsubdivided population of the same total size, as a function of the migration rate (x-axis) and the dominance coefficient. Parameter values are n = 100; N = 30; s = 0.01; u = 10⁻⁵; v = 10⁻⁶; α = 0; and h = 0, 0.05, 0.1, 0.2, 0.3, 0.5, and 1 from bottom to top. (B) Absolute average frequency of A for the same parameter values, h = 0, 0.05, 0.1, 0.2, 0.3, 0.5, and 1 from top to bottom.

Figure 3 shows that decreasing N increases the value of m that minimizes the mean frequency of recessive deleterious mutations. The deterministic approximation (35) predicts that the frequency of A should decrease when Nm decreases up to 40

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Figure 3.—

Average frequency of allele A in a subdivided population under hard selection, relative to its average frequency in a nonsubdivided population of the same total size, as a function of the migration rate (x-axis) and deme size. Solid line, n = 30, N = 100; dashed line, n = 100, N = 30; dotted line, n = 300, N = 10. Other parameters are s = 0.005, h = 0, u = 10⁻⁵, v = 10⁻⁶, and α = 0. Simulation results are squares, n = 30, N = 100; open circles, n = 100, N = 30; solid circles, n = 300, N = 10.

If Nm is lower than this limit value, the frequency of A increases as Nm decreases. Although Equation 35 no longer predicts very well the mean frequency of A when N is small and/or m is large, we found that Equation 40 usually gives correct predictions of the position of the minimum. Equation 40 simplifies to when h = 0, and Nm = ¹/₂ when h = 0.1; it also shows that the frequency of A always increases as Nm decreases for h > ¹/₃.

The effect of population structure on the mutation load is illustrated in Figure 4 ; when h > 0 but < ∼¹/₄, decreasing m has a nonmonotonic effect on the load (the load first decreases and then increases), while when h = 0 or > ∼¹/₄, decreasing m always increases the load. When h is between 0 and ¹/₄, an approximation for the value of Nm that minimizes the load can be obtained from the deterministic approximation (36): the load should be minimal when Nm equals 41

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Figure 4.—

(A) Mutation load in a subdivided population under hard selection, relative to the load in a nonsubdivided population of the same total size, as a function of the migration rate (x-axis) and the dominance coefficient. Parameter values are n = 100; N = 30; s = 0.1; u = 10⁻⁵; v = 10⁻⁶; α = 0; and h = 0 (solid line), 0.05 (dashes), 0.1 (dashes/dots), 0.2 (dashes/double dots), and 0.3 (dots). (B) Absolute load, same parameter values.

For the parameter range that we have tested, we found that population structure can decrease the load substantially only when the number of demes is very large or when selection is moderate to strong. When the number of demes is not very large, population structure only very weakly reduces the load (and only for 0 < h < ¹/₄), while for small values of m the load still increases rapidly as m decreases (Figure 4, left).

When h < ¹/₂, population structure always decreases within-deme inbreeding depression (δ_IS) as illustrated by Figure 5A and always increases heterosis (δ_ST; not shown). Between-deme inbreeding depression (δ_IT) can increase or decrease with the migration rate, depending on the value of h, as can be seen in Figure 5B. When m is small, however, δ_IT always increases as m decreases, for all values of h. From the deterministic approximation (38), one finds that δ_IT has a minimum when Nm equals 42indicating that when h > ¹/₄, δ_IT should always increase as Nm decreases; this corresponds to what we observed for most parameter values after numerical integration over the φ distribution.

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Figure 5.—

(A) Within-deme inbreeding depression (δ_IS) in a subdivided population under hard selection, relative to the inbreeding depression in a nonsubdivided population of the same total size, as a function of the migration rate (x-axis) and the dominance coefficient. Parameter values are n = 100; N = 30; s = 0.01; u = 10⁻⁵; v = 10⁻⁶; α = 0; and h = 0, 0.05, 0.1, 0.2, and 0.3 from bottom to top. (B) Between-deme inbreeding depression (δ_IT) for the same parameter values, h = 0, 0.05, 0.1, 0.2, and 0.3 from bottom to top.

Selfing and population structure:

We observed that, as in an undivided population, self-fertilization decreases the average frequency of the deleterious A allele, for 0 ≤ h ≤ 1 (results not shown). This purging effect of selfing is greatest for recessive mutations (h equal or close to zero). Moreover, the effect of population structure on the average frequency of A depends on the selfing rate α. We have seen in the previous section that, without selfing, population structure decreases the frequency of A when h < ∼¹/₃ (as long as Nm is not too small); selfing decreases this purging effect of population structure and can even reverse it if α is high. This is illustrated by Figure 6 , which, like the previous figures, was obtained by numerical integration over the φ distribution, without making any assumption on N and m. From the deterministic approximation (35), one predicts that for values of α > ²/₃, population structure should always increase the frequency of A, for all values of h; this corresponds to what we observed for most parameter values. Note that Figure 6 represents the mutant average frequency in a subdivided population, relative to its average frequency in a nonsubdivided population of the same total size; as both depend on α, one cannot deduce from the figure what is the absolute effect of α on the average mutant frequency (again, this effect is to decrease the mutant frequency, for all parameter values). The same is true for Figures 7 and 8 .

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Figure 6.—

Average frequency of allele A in a subdivided population under hard selection, relative to its average frequency in a nonsubdivided population of the same total size, as a function of the migration rate (x-axis) and the selfing rate. Parameter values are n = 100; N = 30; s = 0.01; u = 10⁻⁵; v = 10⁻⁶; h = 0; and α = 0, 0.1, 0.2, 0.3, 0.4, 0.5, and 1 from bottom to top. Simulation results are open circles, α = 0; solid circles, α = 0.3; squares, α = 1.

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Figure 7.—

Mutation load in a subdivided population under hard selection, relative to the load in a nonsubdivided population of the same total size, as a function of the migration rate (x-axis) and the selfing rate. Parameter values are n = 100; N = 30; s = 0.1; u = 10⁻⁵; v = 10⁻⁶; h = 0.05; and α = 0, 0.02, 0.05, 0.1, 0.2, and 1 (from bottom to top). Simulation results are open circles, α = 0; solid circles, α = 0.1; squares, α = 1.

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Figure 8.—

(A) Within-deme inbreeding depression (δ_IS) in a subdivided population under hard selection, relative to the inbreeding depression in a nonsubdivided population of the same total size, as a function of the migration rate (x-axis) and the selfing rate. Parameter values are: n = 100; N = 30; s = 0.01; u = 10⁻⁵; v = 10⁻⁶; h = 0; and α = 0, 0.1, 0.2, 0.3, 0.5, 0.7, and 1 from bottom to top. (B) Between-deme inbreeding depression (δ_IT) for the same parameter values, α = 0, 0.1, 0.2, 0.3, 0.5, 0.7, and 1 from bottom to top. Simulation results are open circles, α = 0; solid circles, α = 0.3; squares, α = 1.

When h is significantly greater than zero, selfing reduces the mutation load in nonsubdivided populations (Table 1); we observed the same effect in structured populations (not shown). When h is equal or close to zero, however, selfing does not affect the load in large nonsubdivided populations (L = u), while in structured populations the load is minimal for an intermediate value of α; this can be seen from Equation 36 and is also observed in simulations (not shown). This nonmonotonic effect of α for small h may be due to the fact that selfing increases the efficiency of selection against recessive mutations (by increasing homozygosity), but also increases the effects of drift within demes, since selfing decreases N_e in finite populations.

We have seen that, without selfing, population subdivision decreases the mutation load when h > 0 but < ∼¹/₄, if the number of demes is very large or if selection is moderate to strong. This effect is attenuated by a low rate of selfing, as shown by Figure 7, and is reversed when the rate of selfing is moderate to strong. When h = 0 or >¹/₄, decreasing Nm always increases the load, for all values of α. Overall, we observed that, when α > ∼0.2, population structure always increases the mutation load, for all values of h. Calculations from the deterministic approximation (36) lead to a similar result.

Finally, we found that self-fertilization always decreases δ_IS, δ_IT, and δ_ST when 0 ≤ h ≤ ¹/₂. Selfing does not change the direction of the effect of population structure on δ_IS and δ_ST: decreasing m always decreases δ_IS, for all values of α between zero and one (Figure 8A), and always increases δ_ST (not shown), as long as 0 ≤ h ≤ ¹/₂. For very high selfing rates, however, the effect of population structure on δ_IS is very reduced and disappears completely when α = 1, as shown by Figure 8A. The effect of population structure on δ_IT, however, can change in direction depending on the selfing rate. We have seen that without selfing, decreasing Nm decreases δ_IT when h < ∼¹/₄ (provided that Nm is not too small); Figure 8B shows that this effect is reversed when α is sufficiently high. When h > ¹/₄, however, δ_IT always increases as Nm decreases, for all values of α.

Effects of h and s:

Increasing the dominance coefficient of deleterious mutations (h) always increases the mutation load and decreases the average mutant frequency, the within- and between-deme inbreeding depression, and heterosis (results not shown). Indeed, selection is less effective against recessive mutations than against dominant mutations, and the fitness difference between heterozygotes and mutant homozygotes is greatest for recessive mutations, which explains the effects of h on p, δ_IS, δ_IT, and δ_ST. The effect of h on the mutation load can be explained by the fact that recessive mutants are eliminated most often in the homozygous state, while dominant mutants are eliminated most often in the heterozygous state: therefore it takes fewer genetic deaths to eliminate a given number of recessive mutations than to eliminate the same number of dominant mutations (e.g., Crow and Kimura 1970, p. 301). Although in this article we restricted ourselves to the case 0 ≤ h ≤ 1, the effects of underdominant (h < 0) and overdominant (h > 1) mutations can be addressed by our model. We found similar qualitative results for dominant and overdominant mutations, while the effects of population structure and selfing on the average frequency, inbreeding depression, and heterosis due to underdominant and recessive mutations are qualitatively similar. When h < 0, the load has to be redefined to take into account the fact that heterozygotes have the highest fecundity (segregation load); in that case Whitlock (2002) found that population structure increases the load.

The effects of the coefficient of selection against deleterious mutants (s) are illustrated by Figure 9 . Although mutants become less frequent as s increases, the greater fitness difference between heterozygotes and mutant homozygotes causes inbreeding depression to increase; the effect of s on heterosis, however, is nonmonotonic, a result already obtained by Whitlock et al. (2000). Because our diffusion model does not give accurate results when s > m [i.e., when log(s) > −2 in Figure 9], we could not observe this nonmonotonic effect from the model, but only from the simulations. The effect of s on the mutation load is also nonmonotonic. As s decreases, the average frequency of deleterious mutations increases, but their effect on fitness decreases; in the deterministic regime, these two effects exactly compensate each other, and the load does not depend on s. At low values of s, however, deterministic solutions become less and less accurate, and the average frequency of mutants becomes higher than the deterministic equilibrium, causing the load to increase. In the limit as s tends to zero, deleterious mutants have no effect on fitness and the load equals zero. Figure 9 shows that the value of s that maximizes the load (∼10⁻⁴ in Figure 9) is lower than the value of s that maximizes heterosis (∼10⁻²).

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Figure 9.—

Effects of the strength of selection against deleterious mutants (s) on the average mutant frequency (p), the mutation load (L), within-deme inbreeding depression (δ_IS), and heterosis (δ_ST). The curves correspond to numerical integration over the φ distribution (with hard selection), and the dots correspond to simulation results. Parameter values are n = 100, N = 100, h = 0.01, α = 0, m = 0.01, u = 10⁻⁵, and v = 10⁻⁶.

DISCUSSION

In this article we expressed the mutation load, inbreeding depression, and heterosis in an island model of population structure as functions of the first moments of the frequency distribution of a deleterious allele in the metapopulation. We then used a diffusion model to calculate this distribution. Diffusion methods for island models of population structure use the fact that, as the number of demes tends to infinity, the ancestral lineages of different genes from the same deme either stay in the same deme and coalesce relatively fast or move to a different deme and take a very long time to join again in a deme. This has been described by some authors as a separation of timescales (Ethier and Nagylaki 1980; Wakeley 2003). By neglecting the effect of selection on the average coalescence time within demes, one can then use neutral probabilities of coalescence to obtain expressions for the variance of allele frequencies among demes. This gives good results as long as selection is small relative to allele frequency fluctuations within demes, i.e., approximately when s < m + 1/N (Roze and Rousset 2003). Although the first moments of the probability distribution of the frequency of the deleterious allele have to be obtained by numerical integration, some approximations are possible, the simplest being Equation 33, which also corresponds to the deterministic equilibrium. We found that Equation 33 gives good results as long as h > ∼0.3, or as h < 0.3 and the migration rate m is small. If one then assumes that m is small and that deme size N is large, Equation 33 becomes a simple expression that leads to Equations 35–39. These are the simplest approximations for the mean mutant frequency, mutation load, inbreeding depression, and heterosis that we could obtain. The diffusion method also assumes a high number of demes and weak selection, but these hypotheses are in fact not very restrictive, as long as s < m, as illustrated in Table 3 of Roze and Rousset (2003). When s > m, we found that the solutions obtained by Glémin et al. (2003) are more accurate than ours, if h is not small (indeed the method used by Glémin et al. is not appropriate for small values of h, since they assume that deleterious alleles are present only in the heterozygous stage). This is illustrated by Figure 10 , which shows δ_IS calculated from Equations 3a, 8, and 10a from Glémin et al. (2003), from our model after numerical integration over the φ distribution, and from our approximation (37).

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Figure 10.—

Within-deme inbreeding depression (δ_IS) in a subdivided population under hard selection, as a function of the migration rate m. Solid curves, solution obtained from Equations 3a, 8, and 10a in Glémin et al. (2003); dashed curves, numerical integration over the φ distribution; dotted curves, approximation (40) from this article; dots, simulation results. Parameter values are n = 100, N = 100, s = 0.1, u = 10⁻⁵, v = 10⁻⁶, α = 0, (A) h = 0.3, and (B) h = 0.1.

Whether the diffusion method can be used for other models of population structure remains unclear. Whitlock (2002) assumed that M_δp has the same form in a stepping-stone model as in the island model, but there is no proof of this assumption (see Rousset 2002; Rousset 2004, p. 82). Aware of this problem, Maruyama (1983) derived diffusion approximations by noting that M_δp/V_δp can be computed despite the exact form of M_δp being unknown (see also Rousset 2004, p. 151). Maruyama's argument holds for semidominant mutants but may not be easily extended to other cases, and we made no attempt in this direction. Although Whitlock's diffusion approximations may be reasonable for the parameter values he investigated (Whitlock 2003), a general argument is missing and further investigations would be required.

We obtained simple results about the effects of selfing: the average mutant frequency, within- and between-deme inbreeding depression, and heterosis all decrease as the selfing rate increases; in most cases, the mutation load also decreases as selfing increases, except when deleterious mutations are fully (or almost fully) recessive and population structure is strong, in which cases the load is minimized for an intermediate value of the selfing rate. The effects of population structure are more complicated and depend on the rate of partial selfing. Without selfing, moderate population structure decreases the frequency of deleterious mutants, the mutation load, and between-deme inbreeding depression (δ_IT) when deleterious mutations have a dominance coefficient < ∼0.2–0.3. This purging effect of population structure is decreased by selfing and is reversed when the selfing rate is sufficiently high. When the dominance coefficient of deleterious mutations is >0.2–0.3, population structure always increases the mutant frequency, the mutation load, and between-deme inbreeding depression. Finally, when population structure is very strong, the mutant frequency, mutation load, and between-deme inbreeding depression always increase as population structure increases, for all selfing rates and dominance coefficients. The effects of population structure on within-deme inbreeding depression and heterosis are simpler: δ_IS always decreases, and δ_ST always increases, as the degree of population structure increases.

These results illustrate a double effect of population structure, already shown by Whitlock (2002) in models without selfing: (i) population structure helps to purge recessive deleterious mutations by increasing the average homozygosity, but (ii) also increases the genetic similarity among competing individuals, thereby reducing the effect of selection. The overall effect of population subdivision depends on the relative importance of these two effects. When the selfing rate is low, the first effect prevails when mutations are recessive (h < 0.2–0.3), which explains why the mutant frequency and the mutation load decrease as Nm decreases (unless Nm is very small); with a moderate to high selfing rate, however, selfing becomes more efficient than population structure in increasing homozygosity, and the second effect of population structure, which increases the frequency of deleterious mutants and the mutation load, prevails.

Results about the effects of the selection coefficient s show that weakly deleterious mutations have the strongest effect on the mutation load, while heterosis is maximum for moderately deleterious mutations, and inbreeding depression is maximum for strongly deleterious mutations (Figure 9). Whether heterosis, mutation load, and inbreeding depression are due mainly to weakly, moderately, or strongly deleterious mutations depends critically of course on the probability of occurrence of these different types of mutations, which at present remains poorly known.

Although we assumed throughout the article that migration in the gametic phase was absent, it can be incorporated into the model easily. We modified the life cycle considered in the present article to model a plant life cycle; in this modified model, self-fertilization occurs first (a proportion α of the ovules of a plant being fertilized by pollen produced by the same plant), then pollen migrates at a rate m_p, the 1 − α remaining ovules are fertilized randomly by pollen present in the same deme, and finally seeds migrate at a rate m_s. We found that when m_p and m_s are small and N is large, the deterministic solutions for the average mutant frequency, the mutation load, inbreeding depression, and heterosis are still given by Equations 35–39, m being replaced by m_s + (1 − α)m_p/2. When migration through seeds is very low or absent, and most migration occurs through pollen, the selfing rate α has a nonmonotonic effect on p̅, L, δ_IT, and δ_ST; this is due to the fact that selfing has now the additional effect of decreasing the rate of pollen migration. The effect of α on δ_IS, however, remains unchanged: δ_IS always decreases as α increases.

APPENDIX A

To obtain a and c, we also need to calculate the probabilities that three genes sampled in three different individuals from the same deme all have a common ancestor (d) and that they have no common ancestor (f), in the infinite island model. The recurrence equations are A1 A2 A3 A4 A5 A6which gives for r₀ and r₁, A7 A8

The expressions of a and c are more complicated and not given here for space reasons, but are available by request to the authors.

APPENDIX B

We seek convenient approximations for integrals of the form B1

There is no simple approximation for all conceivable ranges of parameter values, but a useful approximation can be obtained for |σ₁| large, from which other approximations can be derived. Write B2, B3from Equation 13.2.1 in Abramowitz and Stegun (1972)(refered to hereafter as AS), where M(., ., .) is Kummer's confluent hypergeometric function (AS 13.1.2). From this, assuming that |σ₁| is large gives by AS 13.1.5, B4 B5where U(., ., .) is the other Kummer function (AS 13.1.3).

The mean allele frequency is B6with σ₁ = 4N_eS₁, σ₂ = 2N_eS₂, μ = 4N_eu, and ν = 4N_ev. Applying the approximation above, we obtain B7

We assumed that |σ₁| was large to obtain (B7). After comparisons with numerical integrations, we found that (B7) is a good approximation as long as ns is of order 1 or higher. The fit is better as m decreases, because it increases N_e.

Using AS 19.12.4, we obtain from (B7), B8where U(., .) is Whittaker's parabolic cylinder function.

If ≫ μ + , we can use AS 19.8.1, giving B9if we keep only the first term of AS 19.8.1, and B10if we add the second term.

If μ − ≫ , AS 19.9.1 gives B11 B12

In practice, assuming that ns is of order 1, the order of magnitude of depends critically on h and m. increases with h, and when h > 0.3, assuming that the mutation rate is not too large (<10⁻⁴), is large enough for approximation (B9) to give good results. When h < ∼0.3, approximation (B9) can still be used when m is small enough, because increases faster than μ as m decreases; (B10) can be used for larger values of m than (B9). When h is very small (equal or close to zero), (B9) and (B10) work only for small values of m, while (B11) gives good results when m is not small. (B11) does not work well, however, when h is not very small.

The mean of pq over φ is given by B13

Using the same approximation as in (B4), one finds that p̅q̅ is also approximately equal to (B7); therefore the same approximations can be used for p̅ and p̅q̅.