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Inbreeding depression under joint selfing, outcrossing, and asexuality.

Inbreeding depression is thought to be the major genetic factor opposing the evolution of self-fertilization, countering the automatic transmission advantage of selfing first demonstrated by Fisher (1941). The genetic basis for inbreeding depression is now largely held to be recurrent mutation to deleterious recessive alleles, rather than overdominance (for a discussion of the variations on this theme, see Ritland 1996). Theory predicts that, if inbreeding depression is due to deleterious recessive mutations, there should be an inverse relationship between the amount of inbreeding (self-fertilization) and inbreeding depression maintained in plant populations (Lande and Schemske 1985; Charlesworth and Charlesworth 1987). As inbreeding increases, more deleterious recessives are made homozygous, and thus exposed to selection. Inbreeding populations thus "purge" the mutations causing inbreeding depression more efficiently than out-breeding populations, and maintain fewer mutations on average.

This prediction has been largely supported through observations in natural plant populations, but there have been enough exceptions noted to raise doubts about the generality of the theory (Charlesworth et al. 1990; Husband and Schemske 1996). In particular, the discovery of unexpectedly high inbreeding depression in several selfing species has demanded explanation (Agren and Schemske 1993; Eckert and Barrett 1994; Husband and Schemske 1995, 1996). Lower-than-expected inbreeding depression in outcrossers could be attributed to a bottleneck or history of selfing in the recent past. To explain higher-than-expected levels in sellers, however, we must somehow account for the maintenance of inbreeding depression in the face of substantial purging pressure from selection and selfing. Explanations of this observation fall into two classes: those based on the character of the mutations themselves, and those based on some other population-level character.

If mutations are only mildly deleterious, it has been shown that they are much more difficult to purge than lethal mutations, and substantial inbreeding depression may be maintained even in selfing populations (Lande and Schemske 1985; Charlesworth et al. 1990). Lande et al. (1994) showed that even lethal mutations could be very difficult to purge through moderate self-fertilization, if genomic mutation rate, U, and inbreeding depression are sufficiently high. Inbreeding depression is defined as [Mathematical Expression Omitted], where [Mathematical Expression Omitted] is the average fitness of selfed progeny and [Mathematical Expression Omitted] is the average fitness of outcrossed progeny. With very high diploid genomic mutation rates to recessive lethals (U [greater than or equal to] 0.5) and inbreeding depression ([Delta] [greater than or equal to] 0.95), selective interference between loci hinders purging of mutations, even with appreciable selfing rates. Lande et al. (1994) found that in such cases, almost no purging occurred below a "threshold" selfing rate, with a sharp drop-off in load immediately above this selfing rate. This result thus demonstrated that it was possible to maintain very high lethal as well as sublethal mutation loads and inbreeding depression, even in the face of substantial self-fertilization.

The second class of explanation for the maintenance of high inbreeding depression in selfers suggests that more complicated variations on the basic selfing/outcrossing mating system could increase the amount of inbreeding depression maintained in a population. Gynodioecy, polyembryony, and partial asexuality (vegetative reproduction) have all been suggested as candidate systems that could potentially have this effect (Lande et al. 1994). The present study investigates the role of partial asexuality on the dynamics and equilibrium values of inbreeding depression due to recurrent mutation, to determine whether it could provide a reasonable explanation for the anomalous observations.

Several of the examples of strikingly high inbreeding depression have come from species with at least some tendency to reproduce clonally. Epilobium angustifolium (primary selfing rate, s = 0.45; inbreeding depression, [Delta] = 0.95) is a rhizomatous perennial, as is Chinographis japonica (s = 0.89; [Delta] = 0.36) (Maki 1993; Husband and Schemske 1995, 1996). Decodon verticillatus (s = 0.39; [Delta] = 0.607) reproduces asexually through stolons (Eckert and Barrett 1994; Husband and Schemske 1996). Other partially asexual species, such as Opuntia rastrera, have very high estimated inbreeding depression ([Delta] [greater than] 0.5, del Carmen Mandujano et al. 1996). In his monograph on apomixis, Gustafsson reported evidence for inbreeding depression in many other vegetatively reproducing species (Gustafsson 1947a, b). In addition to these observations, there is an intuitive reason for believing that asexual species might harbor more inbreeding depression, if inbreeding depression is due to deleterious recessive mutations. Because plants do not have a separate germline, somatic mutations accumulated over a plant's lifetime may be inherited in the next generation. It has thus been posited that long-lived plants should have substantially higher per-generation mutation rates than short-lived plants (Klekowski and Godfrey 1989). In partially clonal plants, more mitotic cell divisions occur between sexual generations, so that we might expect to see results consistent with effectively higher mutation rates in clonal species.

Vegetative reproduction is widespread in plants. It has been estimated that roughly two-thirds of the shrubs and herbs in Britain are partially asexual (Silvertown 1987). In other northern temperate climates, it has been estimated that [approximately]50% of perennial species reproduce largely through vegetative means, and perhaps as many as 80% of perennial species have at least some capacity for vegetative reproduction (Gustafsson, 1947b). In part because of this prevalence, facultative asexual reproduction is thought to be an extremely successful strategy for plants in general (Abrahamson 1980). Many facultatively asexual species are strongly outcrossing (Ellstrand and Roose 1987), but there is a wide range of observed selfing rates in partially clonal species, with some, such as Iris versicolor, almost entirely self-fertilizing (Kron et al. 1993).

Studies of the evolution of clonal species have focused almost entirely on the question of sexuality versus asexuality, just as studies of the evolution of selfing have focused on the question of selfing versus outcrossing. In this study, we investigate the dynamics of a key parameter of mating-system evolution, inbreeding depression, in the context of partial selfing and partial asexuality. Inbreeding depression was modeled as recurrent mutation to nearly recessive lethals. Asexuality was introduced in two ways: concurrent with sexual reproduction (proportional model), or in a regular alternating pattern with sexual reproduction (cyclical model). We examined how varying degrees of asexuality in these two models affected the levels of inbreeding depression maintained in partially selfing populations.


Basic Model

Inbreeding depression was analyzed using a simplified version of Kondrashov's model of mutation (Kondrashov 1985; Lande et al. 1994). An infinite population undergoes deleterious mutation at an infinite number of unlinked loci. With an infinite number of loci, each new mutation is assumed to occur at a novel locus, and thus be unique in the population. Mutations are identical in their effect on fitness, all having the same dominance and selection coefficients, and are lethal when homozygous. There are then only two possible diploid genotypes at any locus in mature plants, and because all mutations have identical effect, a multilocus genotype may be designated by the number of heterozygous recessive lethals in the genome. The population is then described by the frequency distribution of the number of heterozygous recessive lethals in mature plants.

Each generation, a population undergoes mating, mutation, and selection. In a partially self-fertilizing population, homozygous (lethal) genotypes are produced only through selfed matings, because of the stipulation of the uniqueness of new mutations. Mutation occurs by a Poisson process with a mean of U mutations per diploid genotype per generation. Selection is multiplicative across loci, so that a genotype with i heterozygous recessive lethal mutations has a relative fitness of [Mathematical Expression Omitted], where h is the dominance coefficient and the mean fitness in the population. The number of mutations in the population and their distribution among individuals determine the amount of inbreeding depression, the relative decrement in fitness of selfed versus outcrossed progeny. With inbreeding, the distribution of mutations among loci within individuals is not independent, so that individuals homozygous at one locus are likely to have been the product of inbreeding, and are therefore more likely to be homozygous at other loci (Haldane 1949). This phenomenon of associations of homozygosity among loci is termed identity disequilibrium (Weir and Cockerham 1973), and can be measured in the model by the ratio of the variance to the mean number of recessive lethal mutations.

We ran computer simulations of this model for populations with specified mating systems (proportions of selfing, outcrossing, and asexuality), to characterize equilibrium mutation load and inbreeding depression under different mating systems. When a population closely approached an equilibrium between mutation and selection, the run was stopped and values were recorded for mean number of lethals per genome [Mathematical Expression Omitted], variance in number of mutations per genome (var n), and inbreeding depression ([Delta]).

Proportional Model

Asexuality was incorporated into the basic model in two ways. In the proportional model, in every generation of a particular run the population had the same mating scheme, in which some proportion of an individual's progeny was the product of selfing, some of outcrossing, and the remainder a product of asexual reproduction. This model is appropriate for a plant that consistently both flowers and reproduces vegetatively each generation. For a given run, the proportions of each mating type were constant across generations. There were also no differences in mating proportions among individuals.

In the mating step of the Kondrashov model, progeny genotype distributions were first determined separately for selfing, outcrossing, and asexual matings of the parental generation. These distributions were then weighted by the frequency of each mating type for the overall progeny distribution of genotypes. Equilibrium was defined to occur when the average number of mutations in the current generation differed from that in the hundredth prior generation by less than 0.0001. For a given set of mutational parameters, the model was typically run to equilibrium for 800 populations spread over the range of possible combinations of proportions of selfing (s), outcrossing (t), and asexuality (a), where s + t + a = 1.

Cyclical Model

In the cyclical model, sexual and asexual reproduction did not occur in the same generation. A population experienced some number of purely asexual generations (constant over the course of a run), followed by a single purely sexual generation with a constant proportion of selfing and outcrossing. Such cycles were repeated until the population had reached a pseudoequilibrium, as described below. This scheme approximates a population that reproduces vegetatively over the course of a season, with a single flowering period at the end, or a species that reproduces sexually only when some environmental condition is met, habitually reproducing vegetatively otherwise.

In this model the average number of lethals cycles as the population alternates between sexual and asexual reproduction. Because the fluctuations became very regular, we assessed the approach to a cyclic pseudoequilibrium by comparing the average number of lethals during the current sexual generation with the number during a previous sexual generation, using the same level of sensitivity as the proportional model. Runs were censused during the sexual generation.

For each set of mutational parameters, we considered the effects of varying degrees of asexuality, comparing populations that experienced one generation of sexual reproduction every one, three, five, and 10 generations. In this notation, "every one generation" corresponds to zero asexuality, "every three generations" corresponds to a pattern of two asexual generations followed by one sexual generation, and so on. We also examined populations that reproduced sexually only once every 100 and 1000 generations. We analyzed a range of selfing rates in the sexual generation for each degree of asexuality. In a given run, the selfing rate and the degree of asexuality were constant, and there were no genotypic differences in mating strategy.

Mutational Parameters

The (diploid) genomic mutation rate to recessive lethals has been estimated to be U [approximately equal to] 0.02 per generation in Drosophila (Simmons and Crow 1977). Extrapolation from data on chlorophyll deficiency mutants (reviewed in Klekowski 1992) produces similar estimates of the overall lethal mutation rate for annual plant species. An extrapolation from chlorophyll deficiency mutant data in red mangroves results in an estimate of U = 0.2, consistent with the idea that long-lived plants could, because of their many more mitotic cell divisions per meiotic division, have per-generation mutation rates an order of magnitude higher than annuals (Klekowski and Godfrey 1989; Lande et al. 1994).

For these simulations, we considered mutation rates ranging from U = 0.02 to U = 0.5, assuming for simplicity a single mutation rate for both sexual and asexual generations. Although meiotic mutation rates have been estimated to be several times higher than mitotic mutation rates (Magni and von Borstel 1962; Lindgren 1975), the ultimate effect of this difference on mutation rates in sexual and asexual generations is considerably smaller, because both forms of reproduction undergo many mitotic divisions between generations. With n mitoses per generation the ratio of sexual to asexual per-generation mutation rates is not []/[] (the ratio of meiotic to mitotic mutation rates), but (n + []/[])/(n + 1), which is unlikely to be very large if n [greater than] ([]/[]). The dominance coefficient of mutations in the simulations was generally h = 0.02, in accordance with the Drosophila data on dominance of lethals, but a few runs were done with h = 0.002 and h = 0.05 for comparison.


Proportional Model

For the proportional model, results were expressed as contour plots on a triangular (barycentric) coordinate system. Points within the triangle represent populations with mating proportions (asexual, selfing, outcrossing) described by the perpendicular distances of the point from the three sides of the triangle, with the three vertices representing populations of pure mating type. The bottom side of each figure is the familiar case of zero asexuality, with selfing rates ranging from zero to one. Lines parallel to the s = 0 side represent isoclines of constant [s.sub.abs], the absolute selfing rate in the entire population, [s.sub.abs] = s. In practice, selfing rate in partially asexual populations is measured relative to the number of sexual matings, not the total number of reproductive events. In terms of this relative selfing rate, [s.sub.rel] = s/(s + t), lines radiating from the a = 1 vertex represent isoclines of constant [s.sub.rel].

For low mutation rates (U = 0.02 to U = 0.2; h = 0.02) and correspondingly low inbreeding depression, contour lines of equilibrium inbreeding depression and average number of lethals are nearly parallel to the s = 0 side, corresponding to [s.sub.abs] isoclines [ILLUSTRATION FOR FIGURES 1A, B OMITTED]. The minor deviations of these lines from parallelism, most evident for very high proportions of asexuality, indicate a slight decrease in equilibrium inbreeding depression with increasing asexuality. With respect to [s.sub.rel] isoclines, however, increasing the proportion of asexuality always results in higher average equilibrium inbreeding depression. Identity disequilibrium, measured as the variance to mean ratio of number of recessive lethal mutations in adult plants, generally increases with increasing asexuality, both with respect to relative and absolute selfing rate [ILLUSTRATION FOR FIGURE 1C OMITTED]. The greatest identity disequilibrium occurs in populations with a very high proportion of asexuality. This corresponds roughly to the region of lower-than-expected equilibrium inbreeding depression [ILLUSTRATION FOR FIGURES 1A, B OMITTED].

For high genomic mutation rates (U [greater than] 0.2, h = 0.02), a similar, but more striking, pattern emerges [ILLUSTRATION FOR FIGURES 1D, E, F OMITTED]. At the a = 0 line (no asexuality), a sharp threshold of mutational load and equilibrium inbreeding depression is observed, as in Lande et al. (1994). With increasing asexuality, this threshold disappears, and the reduction in inbreeding depression with selfing becomes more gradual as asexuality increases [ILLUSTRATION FOR FIGURES 1D, E OMITTED]. Variance to mean ratio is highest in regions of high asexuality, and is substantially higher than in the case of low genomic mutation rates and inbreeding depression [ILLUSTRATION FOR FIGURE 1F OMITTED]. The effect of increased asexuality on inbreeding depression and mutational load differs depending upon the relative selfing rate. Above the threshold (relative) selfing rate, the pattern is similar to that for the lower mutation rate [ILLUSTRATION FOR FIGURES 1A, B OMITTED], with inbreeding depression constant with respect to absolute selfing rate, and always increasing with respect to relative selfing rate and an increasing degree of asexuality. Below the threshold selfing rate, however, average number of lethals decreases slightly with increasing asexuality, with respect to both relative and absolute selfing rate [ILLUSTRATION FOR FIGURE 1D OMITTED].

Cyclical Model

In the cyclical model, sexual and asexual reproduction never occurred in the same generation. Selfing rate is thus unambiguously defined as the proportion of selfed matings in the sexual generation. This is equivalent to the relative selfing rate of the proportional model, and to the usual definition of selfing rate in partially asexual plant populations. All values reported for average number of lethals, inbreeding depression, and variance to mean ratio of recessive lethal mutations, are the values experienced during the sexual generation of the pseudoequilibrium cycles.

Results for the cyclical model resemble those in the proportional model. For low genomic mutation rates to recessive lethals, increasing the degree of asexuality increased the average number of lethals and inbreeding depression at equilibrium [ILLUSTRATION FOR FIGURES 2A, B OMITTED]. The variance to mean ratio also increased with asexuality [ILLUSTRATION FOR FIGURE 2C OMITTED].

With a genomic mutation rate high enough to produce a threshold effect, the influence of asexuality again depends upon the selfing rate and its position relative to the threshold selfing rate [ILLUSTRATION FOR FIGURES 2D, E, F OMITTED]. Around and below the threshold rate, increasing the number of asexual generations between sexual generations results in a slight decrease in the equilibrium number of lethals maintained in a population [ILLUSTRATION FOR FIGURE 2D OMITTED]. Asexual reproduction also causes this threshold to become much less pronounced. At high selfing rates, above the threshold selfing rate observed in purely sexual populations, asexuality tends to increase the average number of lethals and inbreeding depression at equilibrium [ILLUSTRATION FOR FIGURES 2D, E OMITTED]. Identity disequilibrium generally increases with increasing asexuality and is again higher than for the U = 0.2 case [ILLUSTRATION FOR FIGURES 2C, F OMITTED].


The results demonstrate that partial asexuality, either cyclical or simultaneous with sexual reproduction, can help maintain high inbreeding depression in highly selfing populations, where the selfing rate is defined as the proportion of sexual matings that are self-fertilizations. The increase in average mutational load and inbreeding depression with increasing asexuality may be attributed to the increase in mutation rate per sexual generation, but that alone is insufficient to explain the overall pattern of inbreeding depression seen. In addition to an increase in mutation rate per sexual generation, which tends to drive the mutational load up, partial asexuality can result in an increase in selection efficiency, which has the opposite effect, driving the average number of mutations down. The case of high genomic mutation rate and high inbreeding depression (U = 0.5, h = 0.02) presents an extreme case that serves to illustrate the interactions of these two basic forces and their dependence upon the selfing rate.

If the only effect of partial asexuality on equilibrium mutational load were mediated through the increase in mutation rate per sexual generation, we would expect to see higher threshold selfing rates as asexuality increases. We found, however, that not only did the threshold rate not increase, the threshold itself disappeared as asexuality increased. In addition, the general tendency for partial asexuality to increase mutational load is contradicted at low selfing rates, around and below the threshold selfing rate for purging of recessive lethals in the purely sexual case. This pattern may be understood by considering the three main forces in this model affecting the number of recessive lethals in a population: mutation itself, purging of lethal homozygotes through selfing, and selection against heterozygotes in adult plants.

Both mutation and selection against heterozygous mature plants occur in every generation, whether sexual or asexual. Purging of homozygous lethals, however, occurs only by selfing (sexual reproduction). The effect of partial asexuality on the equilibrium inbreeding depression depends upon the importance of purging (through selfing), relative to selection against heterozygotes, in determining the number of mutations maintained. For very high mutation rates there is a range of selfing rates (from zero to the threshold rate) for which there is little purging of recessive lethals, and in this range a second effect, an increase in selection efficiency against heterozygotes, is revealed.

Partial self-fertilization generates identity disequilibrium, the association of homozygous genotypes between loci (Haldane 1949). This has the effect of increasing individual variation in fitness, and thus increasing the opportunity for selection (Arnold and Wade 1984). The presence of partial asexuality enhances overall identity disequilibrium, and thus further enhances the efficiency of selection. The increased efficiency of selection against heterozygous mutations with asexuality is responsible for the observed decrease in inbreeding depression below the threshold selfing rate. This effect is still present for higher selfing rates, but because considerable purging of lethal homozygotes also occurs, it is largely swamped by the primary effect of asexuality, a higher mutation rate per sexual generation. The overall result at moderate to high selfing rates is thus an increase in inbreeding depression with increasing asexuality. The pattern is consistent between the proportional and cyclical models. Our theoretical results therefore confirm the possibility of high levels of inbreeding depression in selfing plant species with substantial clonal reproduction.

In some genetic models of the evolution of self-fertilization, inbreeding depression plays a crucial role in determining whether or not selfing will be selectively favored (Lande and Schemske 1985; Charlesworth and Charlesworth 1987; Charlesworth et al. 1990). With respect to the lethal component of inbreeding depression, the primary parameter determining the equilibrium level of inbreeding depression is not [s.sub.rel] (as is usually measured) but [s.sub.abs], the absolute amount of selfing in a population [ILLUSTRATION FOR FIGURE 1B OMITTED]. If inbreeding depression does have a substantial effect on mating system evolution, partial asexuality thus may indirectly affect the evolution of (relative) selfing rates by making the population behave (in terms of inbreeding depression) like one with a lower selfing rate. This would imply that the evolution of self-fertilization should be more difficult in the presence of asexuality.

Clonal plants, if they reproduce sexually at all, are predominantly outcrossing, as are their sexual relatives (Gustafsson 1947a; Silander 1985; Ellstrand and Roose 1987; Dole 1992). Substantial vegetative reproduction and high selfing rate are rarely found in a single population, although there are exceptions, such as I. versicolor (Kron et al. 1993) and some populations of Mimulus guttatus (Dole 1992). Theoretical results of Charlesworth (1980) suggest that if there is a substantial cost of meiosis, asexuality is unlikely to evolve in a selfing population. The results of the present study suggest that there might be an additional genetic constraint on the joint evolution of selfing and vegetative reproduction in the form of increased inbreeding depression (and thus selection against selfing) accompanying asexuality. While it is clearly feasible for selfing species such as Epilobium and Decodon to maintain high levels of inbreeding depression through partial asexuality, it remains a puzzle as to how the combination of high selfing rate and asexuality could have arisen at all.


The authors wish to thank D. Butcher, S. Ratner, D. Schemske, S. Schultz, and J. Willis for discussions. CM thanks J. Kohn for office space, use of computers, and perceptive comments during the writing of this paper. CM was supported by National Institutes of Health grant GM-07413; RL was supported by National Science Foundation grant DEB-9225127.


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Author:Muirhead, Christina A.; Lande, Russell
Date:Oct 1, 1997
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