Abstract
This paper examines the extent to which empirical estimates of inbreeding depression and inter-population heterosis in subdivided populations, as well as the effects of local population size on mean fitness, can be explained in terms of estimates of mutation rates, and the distribution of selection coefficients against deleterious mutations provided by population genomics data. Using results from population genetics models, numerical predictions of the genetic load, inbreeding depression and heterosis were obtained for a broad range of selection coefficients and mutation rates. The models allowed for the possibility of very high mutation rates per nucleotide site, as is sometimes observed for epiallelic mutations. There was fairly good quantitative agreement between the theoretical predictions and empirical estimates of heterosis and the effects of population size on genetic load, on the assumption that the deleterious mutation rate per individual per generation is approximately one, but there was less good agreement for inbreeding depression. Weak selection, of the order of magnitude suggested by population genomic analyses, is required to explain the observed patterns. Possible caveats concerning the applicability of the models are discussed.
1 INTRODUCTION
There is now a substantial body of data on mean population fitness, inbreeding depression and heterosis in subdivided populations, reviewed by Byers & Waller, (1999), Keller & Waller (2002) and Leimu, Mutikainen, Koricheva & Fischer (2006). A number of relevant theoretical investigations have also been carried out (Escobar, Nicot & David, 2008; García-Dorado 2007, 2008; Glémin, Ronfort & Bataillon, 2003; Roze, 2015; Roze & Rousset, 2004; Theodorou & Couvet, 2002; Whitlock, Ingvarsson & Hatfield, 2000). These have shown that the mutational load and inbreeding depression are generally increased by high migration rates and large local population sizes, whereas heterosis in between-population crosses is reduced. While the theoretical expectations are in qualitative agreement with the empirical results, no attempt has been made to determine whether the data quantitatively match theoretical predictions based on estimates of mutation rates and the distribution of fitness effects of new deleterious mutations obtained from genomic data, which have been reviewed by Keightley (2012) and Charlesworth (2015).
Furthermore, the theoretical work has assumed that mutation rates per locus or nucleotide site are small, relative to the strength of selection against deleterious alleles. In addition, the forward mutation rate (from wild-type to mutant) is usually assumed to be much greater than the backward mutation rate (from mutant to wild-type), the latter often being ignored. The first assumption is usually well-justified for mutations affecting protein sequences and strongly constrained functional non-coding sequences. The second assumption applies to the totality of mutations that affect a functional unit such as a coding sequence, provided that selection is sufficiently strong in relation to drift that mutant alleles are close to deterministic equilibrium. This is because there are many ways in which a wild-type sequence can mutate to a sequence with impaired function, but the reversion of a mutated sequence requires a change at the site of the original mutation. However, if we are considering individual nucleotide sites under weak selection, models that include drift and reverse mutation need to be used (Charlesworth, 2013; Kimura, Maruyama & Crow, 1963; Kondrashov, 1995).
The importance of examining the consequences of relaxing these two assumptions is brought out by recent studies of variation in epigenetic marks in plants such as Arabidopsis thaliana, especially methylation at CG sites. These have found rates of gain and loss of marks at individual nucleotide sites that are about 5,000 times higher than the DNA sequence mutation rate, with a strong bias towards loss of methylation (Quadrana & Colot, 2016; Van De Graaf et al., 2015; Vidalis, Zivkovic, Wardenaar, Roquis, Tellier & Johannes, 2016). If fitness is affected by epiallelic variants at numerous sites in the genome, the above assumptions are likely to be violated (Charlesworth & Jain, 2014).
In this paper, theoretical expectations are derived for the mutational load, inbreeding depression and between-population heterosis in subdivided populations, allowing for the possibility of some level of inbreeding, as well as for potentially very high mutation rates per nucleotide site, using an extension of the method described by Charlesworth & Jain (2014). The results are related to data on the relevant parameters obtained from the literature, using parameters of mutation and selection derived from genomic studies. It is shown that there is fairly good quantitative agreement between the theoretical predictions and estimates of heterosis and the effects of local population size on genetic load, but less good agreement for inbreeding depression. The ground is prepared by describing the basic model of a single randomly mating population.
2. THEORETICAL METHODS AND RESULTS
2.1 Basic model of a randomly mating population of infinite size
The standard population genetic model of selection and mutation at a biallelic autosomal locus is assumed (Charlesworth & Charlesworth, 2010, Chap. 4), The relative fitnesses of the genotypes A1A1, A1A2 and A2A2 are 1, 1 − hs and 1 − s; the frequencies of A1 and A2 in a given generation are p and q, respectively. The forward and backward mutation rates (A1 to A2 and A2 to A1, respectively) are u and v. It is convenenient to write u = κv, where κ measures the extent of mutational bias. If κ > 1, mutation is biased towards the production of deleterious alleles; if κ < 1, the bias is toward wild-type alleles.
Ignoring second-order terms in s, u and v, the change in q per generation in a randomly mating, discrete generation population is given by: The mean fitness of the population is: and the genetic load is:
The equilibrium frequency of A2, q*, is found by equating equation (1) to zero. Further details are described in the Supplementary Material, section 1. The corresponding equilibrium load, L*, given by this procedure is always smaller than the low mutation rate approximation, 2u, for non-recessive autosomal mutations (Haldane, 1937), but approaches 2u when hs ≫ (u + v).
Table S1 and Figure S1 show some numerical examples of the dependence of q* and L* on the selection and mutation parameters, which are discussed in the Supplementary Material, section 1. In the next section, the effects of finite population size are examined.
2.2 Stochastic results for a randomly mating population
Here, the load is measured by taking its expected value over the distribution of allele frequencies, following the approach used by Kimura et al. (1963) and Bataillon & Kirkpatrick (2000) for the low mutation rate case. Following Kimura et al. (1963), insertion of equation (2) into the general expression for the stationary probability distribution of allele frequencies at a single locus (Wright, 1937a; Charlesworth & Charlesworth, 2010, Chap. 5) yields an expression for the probability density of q in a randomly mating population with effective population size Ne: where Ne is the effective population size, α = 4Neu, β = 4Nev, and C is a constant of integration that ensures that the integral of ϕ(q) between 0 and 1 is unity.
Equations (S4) – (S7) of section 2 of the Supplementary Material, show how this expression can be used to obtain explicit formulae for the moments about zero of the allele frequency at mutation-selection-drift equilibrium, using infinite series representations of the relevant integrals. This avoids the need to use either numerical integration or the linearized approximation for the effect of selection, given by equations (S10) and (S11) in the Supplementary Material, section 3. Both of these approaches have been employed in past treatments, e.g. Bataillon & Kirkpatrick (2000); Glémin et al. (2003), and Roze & Rousset (2004). For an alternative approach, see García-Dorado (2007, 2008).
Some results for the dependence of the expected load (equation S8) on the selection and mutation parameters are shown in Figure S2. The general picture is very similar to that described by Kimura et al. (1963). There is a critical value of the scaled selection parameter, γ = 2Nes, below which drift and mutation start to overcome selection, leading to a load that is much greater than the deterministic equilibrium value. While high mutation rates tend to reduce the expected load relative to its deterministic value, its absolute value always increases with the mutation rate. As shown below, the same pattern is observed in subdivided populations.
2.3 Analytical results for a subdivided population with random mating within local populations
It is useful to consider first some general results on subdivided populations, which do not assume a specific population structure. While similar results have been obtained before (Escobar et al., 2008; Glémin et al., 2003; Roze, 2015; Whitlock et al., 2000), the simple but general derivation given below does not seem to have been published. Here, inbreeding depression is defined as the difference between the expected mean fitness over all demes and the expected fitness of completely homozygous individuals formed from alleles sampled within a deme. Because the effect of a single locus is very small, this measure is equivalent to the B coefficient of Morton, Crow & Muller (1956). This coefficient is equal to the regression of the natural logarithm of fitness on the inbreeding coefficient of an individual, and is widely used to describe empirical estimates of the effects of inbreeding, e.g. Charlesworth & Charlesworth (1987) and Charlesworth & Charlesworth (2010, Chap. 4). On the assumption of multiplicative fitness effects of multiple independent loci, B is given by the sum of the individual locus values.
For a single locus, whose effect on mean fitness can be assumed to be very small, the expected load suffered by a deme (equation S8) can be written as: where Fs is the inbreeding coefficient that measures population differentiation at a locus subject to mutation and selection, overbars indicate expectations over demes, and the subscript l indicates that Ll is the value for a single locus or nucleotide site.
This equation brings out the important fact that the mean load is affected by both the effects of drift on the mean of q over all demes, and on the extent of variation in q among demes when h ≠ ½.
The corresponding measure of the expected inbreeding depression is:
This shows that population subdivision tends to reduce the mean inbreeding depression when h < ½, even if the mean frequencies of mutant alleles are unchanged from the random mating case.
The value of Bl under classical mutation-selection equilibrium with u ≪ hs is 2u[1/(2h) − 1], provided that h > 0 (Charlesworth & Charlesworth, 1987). It is helpful to use the quantity Brel = Bl /2u as an estimate of the value of the sum of the Bl over loci relative to the diploid genome-wide deleterious mutation rate, U, which is equal to the sum of the values of 2u over all loci. A relative mean load, Lrel = Ll/(2u), can be defined in the same way.
Similarly, the expected between-deme heterosis (Hl) caused by a single locus is given by the expected difference between the fitnesses of random matings among individuals from different demes and random matings within demes (Glémin et al., 2003). This is given by: where Vq is the variance among demes in allele frequency at the selected locus. Again, use of Hrel = Hl /(2u) allows the sum of Hl values over loci to be compared with U.
These expressions bring out clearly that the extent of population subdivision, as measured by Fs, has complementary effects on Bl and Hl. Fs is expected to be smaller than the genealogical measure of subdivision, FST, which assumes neutral transmission (Glémin et al., 2003). However, since FST is relatively easy to measure using genetic markers, it can be used as a rough guide as what to expect from equations (6) and (7) (Charlesworth & Charlesworth, 2010, pp.361-362). The numerical examples described below for the infinite island model suggest that the neutral approximation works well when s ≪ m, as is predicted by equation (S12).
Further results can most be easily obtained using Wright’s island model of population structure with an infinite number of demes (Wright, 1937b), which assumes that the metapopulation is composed of a large number of randomly mating demes, each with effective size Ne. Each deme receives a constant fraction m of its gene pool from the population as a whole, with an expected frequency qt of allele A2 among the migrants that is equal to the mean allele frequency for the whole metapopulation, which is assumed to be fixed. This corresponds to soft selection in the sense of (Whitlock et al., 2000).
The stationary distribution of allele frequencies among demes under the infinite island model (Wright, 1937b) can be found from the analogue of equation (4): where M is the scaled migration parameter, 4Nem, and pt= 1 − qt.
However, in general qt cannot be equated to the deterministic equilibrium value, q*. It can be found by iteratively determining the value of qt that corresponds to the mean of q over the distribution given by equation (8), using a simple extension of the method described in the Supplementary Material, section 2. Linearized approximations are provided by the equivalents of equations (S10) and (S11).
2.4 Numerical results for a subdivided population with random mating within local populations
The results described here assume weak selection, with s = 2.5 × 10−3. There are two reasons for this; first, unless deme sizes are extremely small (≤ 100), the effects of drift will be minor when selection coefficients are an order of magnitude or more greater than this (Kimura et al., 1963). Second, recent population genomic studies of Drosophila suggest that there is a wide distribution of selection coefficients against deleterious nonsynonymous mutations and mutations in functional non-coding sequences, with a mean selection coefficient (hs) of the order of 10−3 for nonsynonymous mutations, and a much smaller value for mutations in functional non-coding sequences (Campos, Zhao & Charlesworth, 2017; Kousathanas & Keightley, 2013). The outcrossing flowering plant, Capsella grandiflora, has a similar estimated distribution of the fitness effects of nonsynonymous mutations to that in Drosophila (Slotte, Foxe, Hazzouri & Wright, 2010). With the highly left-skewed distributions suggested by these analyses, most deleterious mutations will have considerably smaller s values than the mean. To be conservative in the sense of underestimating the effects of drift, s = 2.5 × 10−3 was chosen for the numerical analyses.
Table 1 shows some examples of the effects of varying the deme size, mutation rate and scaled migration rate on the exact and approximate values of the expectations of the parameters of interest. As described in section 2.3, these are divided by the diploid mutation rate, 2u, as indicated by the subscript rel. There are several points of interest. First, for the smaller deme size the exact expected load is considerably higher than 2u for the lower pair of values of the scaled migration rate M, reflecting the effect of drift in causing deleterious alleles to drift to high frequencies when 2Nes is of order 1 or less. This effect is not seen with the linear approximation, because this assumes that the mean of q is equal to the deterministic equilibrium value, q*.
With M = 10, the load approaches the classical deterministic value, 2u, and is even slightly lower than this when h = 0.2, reflecting a weak effect of purging of partially recessive alleles from large but finite populations (Wang, Hill, Charlesworth & Charlesworth, 1999). As is also the case for a single population (sections 2.1 and 2.2), the higher mutation rate is associated with smaller values of the load relative to 2u, especially with mutational bias towards favoured alleles, but the absolute mean load is much higher than with the lower mutation rate, even with Ne = 100 and M = 0.1. When h = 0.2, the linearized approximation of equation (S12) gives quite a good fit to Lrel, Brel and Hrel for the cases with M ≥ 1 and Ne = 100; the fit is good for all three values of M when Ne = 1000, reflecting the relatively small effect of drift in this case. For M = 0.1, there are substantial discrepancies between the approximate and exact values, reflecting the large departures from q*.
Second, as expected from equations (6) and (7), the values of Brel and Hrel for h = 0.2 respond in opposite directions to changes in Ne and M, with the level of heterosis being similar to, or even greater than, 2u when deme size is small and M ≤ 1, but is much less than 2u when deme size when deme size is large or M ≥ 10. Brel for h = 0.2 is mostly well below its deterministic value of 1.5 for the smaller pair of values of M, but approaches this value when M = 10. Fs was found to be close to the genealogical value, FST = 1/(1 + M), for most of the examples in Table 1, except for Ne = 1000 and M < 10 as well as 2Nes ≫ M.
There could thus be a substantial expected level of heterosis in interpopulation crosses, even with moderately large Ne or M, when selection is of the strength considered here. With U = 1, the sum of the H values over all sites with the selection and mutation parameters in Table 1 would be approximately 0.60 with h = 0.2, Ne =100 and M = 1, and approximately 0.10 even with Ne =1000 and M = 10.
Finally, the high mutation rate cases in Table 1 all give values of Lrel, Brel and Hrel that are smaller than the corresponding values with low mutation rates; the differences are more marked for Lrel than the other variables. However, the absolute values of all three variables are always much greater with the high mutation rate. Highly mutable, non-neutral epimutations could, therefore, cause high levels of mutational load, inbreeding depression and heterosis.
2.5 Analytical and numerical results for subdivided populations with inbreeding
As discussed in section 3 below, much of the evidence for inbreeding depression and interpopulation heterosis in subdivided populations comes from populations with a mixture of inbreeding and outbreeding. It is therefore of interest to consider the implications of inbreeding for the above results. If selection is weak, the evolutionary dynamics of an inbreeding population can be approximated to the order of second-order terms in s by using the neutral value of the within-population inbreeding coefficient, FIS (Charlesworth, Nordborg & Charlesworth, 1997). The modifications to the relevant equations are given in section 5 of the Supplementary Material, equations (S13) – (S17).
Only the case of a subdivided population will be considered in detail here; the results for large M provide a picture of what is expected for a single large population. If Bl for a single locus in a partially inbreeding population is defined as the difference between the mean fitness of random matings within demes and the mean fitness of fully homozygous individuals formed by sampling alleles within demes, and Hl as the difference in mean fitness between random matings between demes and random matings within demes, no changes to equations (6) and (7) are needed.
Table 2 shows results for Ne = 100 and FIS = 0.9, which can be compared with the results for the same values of Ne and the selection, mutation and migration parameters in Table 1. (Ne with inbreeding is obtained by dividing Ne for the corresponding random mating case by (1 + FIS) (Laporte & Charlesworth, 2002; Pollak, 1987), so that this example corresponds to a larger actual deme size than in Table 1.) It has long been known that the greater effectiveness of selection on a single locus with inbreeding causes the genetic load due to selection against non-recessive deleterious mutations to approach u instead of 2u in a large population when mutation rates are low (Charlesworth & Charlesworth, 1987; Crow 1970; Lande & Schemske. 1985). The mean genetic load in an inbreeding metapopulation is thus expected to be reduced by inbreeding (Glémin et al., 2003; Roze & Rousset, 2004), as can be seen by comparing Tables 1 and 2. With M = 10, both the exact and approximate values of the mean load approach u, even with the high mutation rate, and are substantially less than the values for the corresponding random mating cases, including the examples with lower M values.
For Brel and Hrel, the patterns are similar to those found with random mating but with substantially smaller values, except that the exact value of Brel with h = 0.2 does not change greatly with changes in M; instead, it remains quite close to the value given by substituting the deterministic equilibrium value of q from equation (S15) into equation (6) with FIS = 0 (approximately 0.33, as opposed to 1.5 with random mating). This appears to be due to a sharp increase in the mean of q as M decreases, which compensates for the decline in 1 − Fs. Consistent with this interpretation, the linearized approximations for Brel and Hrel perform poorly with M < 10. The approximate constancy of Brel with changing M is also found with Ne = 1000 and h = 0.2. In this case, the mean allele frequency changes little with M, while Fs is held to a low level by selection, and the linearized approximation works well for all M values; Hrel is always small (approximately 0.05 for M = 0.1, and 0.02 for M = 1).
3 EMPIRICAL EVIDENCE ON INBREEDING DEPRESSION AND HETEROSIS IN SUBDIVIDED POPULATIONS
3.1 Plant populations
Table S2 of the Supplementary Material shows some empirical estimates of the mean inbreeding depression and heterosis for flowering plants with subdivided populations. The studies concerned were selected on the basis that estimates of lifetime fitness were available, rather than single fitness components, and that data on multiple local populations were collected. Overall, both B and H are usually modest in size. The largest B values in the table are 2.52 ± 0.60 and 3.32 ± 0.82 (where ± indicates the standard error), for self-incompatible populations of Arabidopsis lyrata (Oakley, Spoelhof & Schemske, 2015) and predominantly outcrossing populations of Sabatia angularis (Spigler, Theodorou & Chang, 2017), respectively. These values are considerably larger than the value of 1.5 expected with h = 0.2 and U = 1 for deterministic equilibrium and outcrossing, but the large standard errors means that it is not possible to tell whether they are too large to be explained on this basis. Strongly selected deleterious mutations with small h values contribute substantially to inbreeding depression in large outcrossing populations (Charlesworth & Charlesworth, 1987; Simmons & Crow, 1977), although they are likely to be purged from highly inbred or very small populations (Lande & Schemske, 1985; Wang et al., 1999). They will not, however, contribute much to heterosis between populations, since their frequencies are not greatly affected by drift (see section 4.3).
The largest values of H in the table are 1.20 ± 0.47 for small populations of Hypericum cumulicola (Oakley & Winn, 2012) and 1.43 ± 0.47 for partially selfing populations of Arabidopsis lyrata (Oakley, et al., 2015). FST among the small populations of H. cumulicola was 0.74 (from Table S1 of Oakley & Winn [2012]), and 0.78 among partially selfing populations of A. lyrata (Mable & Adam, 2007). Patterns of variability at molecular markers indicate that both sets of populations are partially inbred (mean FIS of 0.74 and 0.44 for small populations of H. cumulicola and partially selfing A. lyrata, respectively).
3.2 Animal populations
Relevant data where both B and H were measured for net fitness are less abundant for animal populations, despite the fact that studies showing heterosis in interpopulation crosses were pioneered in Drosophila pseudoobscura, using data on recessive lethals (Wright, Dobzhansky & Hovanitz, 1942) and several fitness components (Vetukhiv, 1953, 1956, 1957). However, outbreeding depression, reflecting the accumulation of Dobzhansky-Muller genetic incompatibilities between local populations, is often observed in animals, especially in hermaphrodite species with high frequencies of self-fertilization (Dolgin, Charlesworth, Baird & Cutter, 2007; Escobar et al., 2008). This makes it hard to assess the extent of heterosis.
The most informative study is probably that of Lohr & Haag (2015), on groups of populations of Daphnia magna inhabiting both small and large ponds. Their results on the net reproductive output per individual under laboratory conditions provide a measure of net fitness. The small ponds had B and H values of 0.13 ± 0.38 and 0.96 ± 0.26, respectively; the corresponding values for large ponds were 1.69 ± 0.31 and – 0.30 ± 0.30. Previous work showed that FST for molecular markers was considerably higher among small than large populations (0.70 versus 0.38 for microsatellites), with much lower within-population diversity in the small populations (Walser & Haag, 2012).
3.3 Small versus large populations
Several of the plant studies also allowed groups of local populations with small census sizes to be contrasted with groups with larger size, or partially self-fertilizing populations to be contrasted with outcrossing populations. If smaller population size and a greater extent of selfing are both associated with reduced Ne and M, as is likely to be the case (Charlesworth, 2003), H should be larger for the small/partially inbreeding populations than the large/outcrossing ones, with the opposite pattern for B. A difficulty with this prediction is that increased inbreeding, with its purging effects, causes a reduction in H as well as B, which could dilute or even reverse the effect of inbreeding on Ne and M. However, even the self-incompatible populations of A. lyrata have substantial FIS values, with a mean of 0.20 (Mable & Adam, 2007), so this difficulty is probably not as severe as might have been expected.
If the Daphnia results are included, and only the Willi (2013) study is used for A. lyrata (to avoid pseudoreplication), five independent contrasts are available for H and four for B. All five of the H differences are in the direction of a greater value for small/selfing populations, with a mean difference of 0.70 ± 0.23 (P = 0.031 on a sign test, or 0.019 on a one-tailed paired t-test, t = 3.06). But of the four contrasts for B, one has a greater value for small/selfing populations (A. lyrata), and the mean contrast between large/outcrossing and small/selfing populations is 0.70 ± 0.42, t = 1.65, P > 0.10. Given that several of these studies reported significant differences in their measures of heterosis between small/selfing and large/outcrossing populations, it seems fairly clear that there is indeed a significant effect for H in this direction; the situation with B is less clear. This is perhaps not unexpected, since the relative effects of differences in M and Ne are more marked for H than B (Tables 1 and 2).
A similar question can be asked about the genetic load. While it is not feasible to estimate the absolute value of the load, because the fitness of the hypothetical mutation-free genotype is unknown, it is possible to compare the mean fitnesses of small and large populations, or selfing and outcrossing populations. The natural logarithm of the ratio of mean fitnesses reflects the difference in the sums of the Ll over all loci between the two groups, under the multiplicative fitness assumption. In the studies listed in Table S2, and in many other studies (Leimu et al., 2006), the individuals whose fitnesses are estimated were the product of random matings among parents sampled from natural populations, and so did not necessarily represent the fitnesses of individuals produced by the natural mating system. This means that the corresponding load for a single locus is described by equation (5), with the proviso that the mean q is the value expected under the natural mating system, not random mating.
The expectation that the load is lower for large than small populations holds true for this measure, consistent with the results of a large meta-analysis (Leimu et al., 2006). The expectation for a comparison between selfing and outcrossing populations is less clear, since selfing is expected to reduce the mutational load caused by partially recessive mutations (Charlesworth & Charlesworth, 1987; Lande & Schemske, 1985), whereas increased drift due to smaller effective population size and reduced migration has the opposite effect. For the cases in Table S2, a significantly positive correlation between population size and fitness (P < 0.05) was found by Paland & Schmid (2003); for the data of Oakley & Winn (2012) on H. cumulicola and Lohr & Haag (2015) on D. magna, the differences in log mean fitness between large and small populations were 1.13 ± 0.51 and 1.22 ± 0.31, respectively, indicating a clear effect of population size in the expected direction.
For the outcrossing versus selfing comparisons in Table S2, the corresponding load differences were 0.50 ± 0.50 (Busch, 2006), 1.25 ± 0.29 (Willi, 2013), and 0.96 ± 0.53 (Oakley et al., 2015) (the last two values are both for A. lyrata). This suggests that the effect of differences in levels of inbreeding between the outcrossing and selfing populations, which are relatively modest in the case of A. lyrata, are outweighed by difference in effective deme size and/or level of population subdivision.
4 DISCUSSION
4.1 The relation between the mutation rate and the genetic load
The theoretical results described in section 2 show that, when the scaled selection coefficient (γ = 2Nes) is ≫ 1 in a single population or in an island metapopulation with a scaled migration rate M = 4Nem ≫ 1, the expected genetic load for a single locus at mutation-selection-drift equilibrium is close to the deterministic value, consistent with previous studies that assumed low mutation rates, e.g. Kimura et al. (1963). The load strongly depends on the mutation rate even when the classic Haldane (1937) formula (Ll = 2u) is not exact.
Dr Frank Johannes has suggested to me that a high rate of reversion from deleterious to beneficial alleles may result in only a weak dependence of the load on the mutation rate, but this does not appear to be case when the results with κ = 2 are compared with those with κ = 0.5 in Tables 1, 2 and S1. Although a high rate of reversion from mutant to wild-type somewhat reduces the load, it does not abolish its dependence on the mutation rate (see the parts of the Table with γ ≫ 1). In this area of parameter space, selection to reduce the mutation rate would be expected if M or Ne, as well as the total genomic mutation rate, are sufficiently large (Drake, Charlesworth, Charlesworth & Crow 1998; Sniegowski, 2000; Lynch, Ackerman, Gout, Long, Sung, Thomas & Foster, 2016). This follows from the fact that, in a randomly mating population with free recombination, the selection coefficient on a modifier of the mutation rate is approximately equal to the product of the mean selection coefficient on the deleterious mutations themselves, and the change in the genome-wide deleterious mutation rate (δU) caused by the modifier. In a completely selfing population, the selection coefficient is approximately equal to 0.5 δU (Drake et al., 1998). δU will tend to be larger, the larger U, leading to stronger selection on the mutation rate.
The picture is very different for the stochastic regime with γ = 2Nes ≤ 1, or M < 1 where the fact that the deleterious allele has an appreciable probability of being at a high frequency means that the load is largely determined by the strength of selection, and is only weakly affected by U (see the parts of the tables with γ ≤ 1). Here, selection on modifiers that reduce the mutation rate is likely to be ineffective in the face of drift (Lynch et al., 2016).
These considerations suggest that the methylation status of CG sites in A. thaliana, with its high epimutation rate (Van De Graaf et al., 2015), is probably close to being selectively neutral with respect to purifying selection at most such sites, otherwise selection should have reduced the mutation rate to a much lower level. This is consistent with the results of a population survey where the site frequency spectrum of CG epialleles was used to estimate the intensity of selection, and no significant departure from neutrality was detected (Vidalis et al., 2016). It is, of course, possible that a subset of these sites are functionally significant, and contribute to genetic load and interpopulation heterosis.
Another possibility is that the functional consequences of these epimutations relate to quantitative traits under stabilizing selection; there is experimental evidence that quantitative trait variation in A. thaliana can be caused by epimutations (Quadrana & Colot, 2016). In this case, the relevant contribution to the genetic load would come from the epigenetic component of the genetic variance in the trait, multiplied by the factor S that determines the strength of selection on the trait as a whole – see equations (2) of Charlesworth (2013). Under a balance between stabilizing selection and mutation, the magnitude of this component in a highly selfing species like A. thaliana depends on the mutational variance, Vm (Charlesworth & Charlesworth, 1995; Lande & Porcher, 2015). Empirical estimates of Vm for a range of quantitative traits are available from the literature (Halligan & Keightley, 2009); these include potential contributions from epigenetic as well as genetic variants. These estimates are surprisingly high compared with what might be expected from DNA sequence mutation rates if each trait were controlled by independent sets of genes (Johnson & Barton, 2005), implying either a high degree of pleiotropy across traits or a substantial contribution from epialleles to quantitative trait variability. Further empirical research is needed to discriminate among these hypotheses.
4.2 General patterns expected from the theoretical predictions
Equations (6) and (7), which apply to both randomly mating and partially inbreeding populations, bring out the complementary relations between the measures of inbreeding depression and heterosis in a metapopulation. For a given mean frequency of deleterious alleles in the metapopulation, increased differentiation among demes as measured by Fs increases Hl and reduces Bl, provided that the dominance coefficient h is less than one-half. An increase in mean q due to drift tends to increase both Hl and Bl, as long as it remains less than one-half. If deme sizes or migration rates are so small that mean q is much greater than one-half, Hl and Bl can be reduced compared with large Nem values; this is more likely to affect Bl, since the associated increase in Fs works against Bl but in favor of Hl. With semidominance (h = ½), the mean load given by equation (5) (and its modification for an inbreeding population, equation S17) is independent of Fs, and always increases with mean q. With h < ½, the mean load increases with both mean q and Fs.
The numerical results described in sections 2.4 and 2.5, where the load parameters are expressed relative to the diploid mutation rate (Tables 1 and 2), suggest that the main effect of Nem on these measures for weakly selected mutations is due to changes in Fs rather than mean q, except when M ≤ 1, since the predictions from the linearized approximation for Fs (equation S12), which assumes that mean q is equal to the deterministic equilibrium value q*, perform quite well for M > 1 unless the mutation rate is very high.
However, unless h approaches zero, Lrel > 1 and Brel ≫ 1 require an increase in mean q above q*, unless h and FIS are close to zero. Even with small h, it is likely that Hrel remains < 1 without an increase in mean q, as described in section 5 of the Supplementary Material. As expected from these considerations, and in line with previous theoretical results (Glémin et al., 2003; Roze & Rousset, 2004), Table 1 shows that smaller Ne and M generally lead to larger Lrel and Hrel, and smaller Brel. Inbreeding within demes leads to lower values of Lrel, Hrel and Brel compared to the same Ne and M with random mating.
4.3 Relations between theoretical predictions and empirical results
On the assumption of multiplicative fitnesses, the values of the load, inbreeding depression and heterosis contributed by deleterious mutations across the whole genome can be roughly predicted by the products U Lrel, U Brel and U Hrel where U is the total diploid genomic mutation rate to deleterious mutations. For Drosophila, data on DNA sequence mutation rates and levels of selective constraint imply that U for weakly selected mutations is approximately one (Keightley, 2012; Charlesworth, 2015). Given that the number of genes in the genomes of flowering plants such as Arabidopsis is approximately twice as large as in Drosophila (www.biology-pages.info/G/GenomSizes.html) and that the mutation rates per basepair in A. thaliana and D. melanogaster are similar (Keightley et al., 2014; Ossowski et al., 2010), it seems safe to assume that U for plants like Arabidopsis is at least one, and may be much larger. The results in Tables 1 and 2 should thus provide a rough guide as to what parameter values to expect for natural populations of flowering plants, when contrasting populations with complete outbreeding with those with a high level of inbreeding. The empirical results summarised in Table S2 are broadly consistent with the range of parameter values in these tables for U of order 1.
The procedures described in sections 2 and 3 of the Supplementary Material can be use to assess the extent to which the data on small and large demes can be explained by the mutation-selection-drift model. Predictions were generated for a wide range of selection coefficients, from s = 2.5 × 10−5 to s = 0.05 and for h = 0.2, which cover the range for deleterious nonsynonymous mutations estimated from population genomics data. These were applied to the infinite island model, using estimates of FIS and FST from molecular marker data for large and small populations of Hypericum cumulicola (Oakley & Winn, 2012), and FST for Daphnia magna, where there is no statistically significant evidence for local inbreeding (Lohr & Haag, 2015). The scaled migration rate, M, was estimated from (1 − FST)/FST. The Ne values for small and large populations were arbitrarily set to 100 and 1000, respectively.
Figure 1 shows the resulting plots of the predicted difference in mean relative load between small and large populations (ΔLrel) against s, as well as the predicted Brel and Hrel values for small and large populations. Overall, the predictions are not very sensitive to s until it becomes very large compared with the mean value suggested by population genomics analyses of Drosophila, which is approximately 10−3 for nonsynonymous mutations (Campos et al., 2017), in which case both ΔLrel and Hrel start to decline rapidly.
In these two examples, ΔLrel with s = 5 × 10−4 was 1.07 for H. cumulicola and 1.68 for D. magna, with empirical values of ΔL of 1.13 ± 0.51 for H. cumulicola and 1.22 ± 0.31 for D. magna. Similar results apply to Hrel (the differences between small and large populations were 0.41 for H. cumulicola and 1.08 for D. magna), compared with observed H values of 0.50 ± 0.55 and 1.26 ± 0.39, respectively. However, the magnitudes of the observed differences in B between small populations and large populations of H. cumulicola (− 1.17 ± 0.55) and for D. magna (− 1.55 ± 0.49) are much larger than those of the differences in Brel ( − 0.03 for H. cumulicola and – 0.19 for D. magna). This reflects a large (but non-significantly) negative B value for the small populations of H. cumulicola and a near zero value for small populations of D. magna, compared with Brel values of 0.32 and 0.68 for the respective small populations. The theoretical results may thus overestimate B for small populations, possibly because the island model does not capture bottleneck effects associated with extinction-recolonization events (see section 4.5).
Figure 2 shows the dependence of the results on the dominance coefficient for examples with weak and strong selection, using the demographic parameters for Daphnia, since dominance will have the greatest effect when there is no inbreeding. It can been see that a high degree of recessivity is necessary to obtain large Hrel values when selection is strong. With weak selection, there is a nearly linear dependence of Brel and Hrel on h, whereas ΔLrel is almost independent of h. With strong selection, ΔLrel is always close to zero, and in fact is slightly negative for h values close to 0.1, reflecting the effect of purging.
The high standard errors of the estimates of B and H, especially B, in many of the studies summarised in Table S1 makes quantitative tests for most individual cases unreliable, especially as estimates of FST and FIS are not always available. The analysis of the coefficients of variation of L, B and H in section 6 of the Supplementary Material suggests that variance in allele frequencies between demes is not a major source of the large standard errors, which must therefore arise from experimental error and environmental variation. This suggests that an intensive study of a favourable system, resulting in more precise parameter estimates, would provide a rigorous test of the theory. Overall, however, it seems clear that weak selection, of the magnitude of that inferred from population genomics data, together with a substantial degree of recessivity of deleterious mutations, is required to explain the data on ΔL and H, on the hypothesis of purely mutational load.
4.5 Caveats and complications
There are several important caveats concerning attempts to interpret the data in terms of the simple models used here, which mean that the quantitative predictions should be regarded with caution. First, the assumption of statistical equilibrium for drift, mutation and selection under an island model of migration is questionable, since species with small population sizes are likely to be subject to bottlenecks and extinction-recolonization processes. Relatively few attempts to include such complications in models of inbreeding depression and heterosis have been made, with the recent exception of Spigler et al. (2017).
It should be noted, however, that a set of populations subject to recent bottlenecks is likely to experience effects that are similar to, but less severe than, making an inbred line, with effects predominantly on Fs rather than mean q (Balick, Do, Cassa, Reich & Sunyaev, 2015; Simons, Turchin, Pritchard & Sella, 2014). B will thus be greatly reduced, because of loss of variability. H will be increased, but only to an extent that corresponds to the inbreeding depression in the ancestral population, since there is no time for an accumulation of deleterious mutations at a high frequency that is responsible for the high values seen for the small M values in Tables 1 and 2. There is, therefore, no reason to expect recent bottlenecks to result in much higher H values than those described here. Recurrent bottlenecks associated with a long history of extinction and recolonization, with the associated very high FST values (Pannell & Charlesworth, 1999; Charlesworth & Charlesworth, 2010, Chap. 7), might have such an effect of H, but there is no evidence for extremely high FST values in the examples analysed here.
Second, the assumption of independence among loci used here is likely to be violated in finite inbreeding and subdivided populations, due to associations among loci created by linkage disequilibrium and identity disequilibrium, resulting in Hill-Robertson interference among selected loci (Felsenstein, 1974). Only modest effects of such interference among slightly deleterious mutations on levels of inbreeding depression in partially selfing populations have been found for multiplicative fitness models with dominance coefficients of the magnitude assumed here (Bersabé, Caballero, Pérez-Figuereoa, & García-Dorado, 2016; Charlesworth, Morgan & Charlesworth, 1992; Kamran-Disfani & Agrawal, 2014; Roze, 2015), unless the population size is very small. Larger effects of interference may occur for the small dominance coefficients (h ≤ 0.02) for severely deleterious or recessive lethal mutation (Kelly, 2007; Lande, 1994; Lande & Porcher, 2017; Porcher & Lande, 2016), leading to much higher frequencies of deleterious mutations than expected in the absence of interference, but the low values of B in most studies in Table S2 suggest that the contributions from such mutations are minor. Indeed, they are likely to be purged from partially inbreeding populations, unless mutation rates are very high (Lande & Schemske, 1985; Kelly, 2007). Lack of independence caused by synergistic epistasis among slightly deleterious mutations appears to somewhat reduce the effect of inbreeding on the level of inbreeding depression (Charlesworth, Charlesworth & Morgan, 1991), especially with very high rates of selfing, but its effect on heterosis has not been investigated.
The consequences of interference for B and H in subdivided populations have only just started to be investigated by rigorous models (Roze, 2015); in this study, interference appeared to have modest effects of increasing B and reducing H, mainly by reducing Fs. However, simulations that incorporate distributions of selection coefficients that are consistent with the population genomics estimates are currently lacking, so that it is currently unclear whether interference alone could explain the discrepancies between observations and theory for B that were described above.
Finally, the possible contributions of strongly selected deleterious mutations and variability maintained by heterozygote advantage have been ignored. As shown above, deleterious mutations with selection coefficients much greater than 10−3 are unlikely to contribute significantly to H and ΔL in a subdivided population, unless deme size is extremely small with a very high level of population subdivision. Heterozygote advantage can be studied by a straightforward extensions of the methods employed here, and its consequences for inbreeding depression in a subdivided population has been investigated by Whitlock (2002).
His results are extended to predictions of H with random mating within demes in the Supplementary Material, section 8. Equation (S21) shows that loci under balancing selection can contribute to heterosis. With weak selection, H is approximately equal to the production of FST and the equilibrium genetic load under balancing selection. Current evidence from population genomic surveys suggest that loci maintained by heterozygote advantage and other forms of balancing selection are sparsely distributed across the genome (Charlesworth 2006; Gao, Przeworski & Sella 2015; Siewert & Voight, 2017); the intensity and mode of selection at such loci is largely unknown. Indirect evidence suggests, however, that balancing selection contributes significantly to genetic variation in fitness in Drosophila (Charlesworth, 2015) and to inbreeding depression (Charlesworth & Charlesworth, 1999), and so could be partly responsible for the observed values of B and H.
Data accessibility
No new data were generated for this paper. Computer code and output files will be deposited in Dryad on acceptance.
Author contributions
Brian Charlesworth designed and conducted the research, and wrote the manuscript.
For the other studies, the means were obtained by log-transforming the values of mean provided in tables or figures, and the standard errors were estimated from the ratios of the untransformed standard errors to the untransformed means.
Acknowledgements
I thank Diala Abu Aswad and Deborah Charlesworth for their comments on a previous version of this manuscript.