SEARCH

SEARCH BY CITATION

Keywords:

  • cost of males;
  • cost of sex;
  • evolution of sex;
  • Muller's ratchet;
  • very slightly deleterious mutations

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

The frozen niche variation hypothesis proposes that asexual clones exploit a fraction of a total resource niche available to the sexual population from which they arise. Differences in niche breadth may allow a period of coexistence between a sexual population and the faster reproducing asexual clones. Here, we model the longer term threat to the persistence of the sexual population from an accumulation of clonal diversity, balanced by the cost to the asexual population resulting from a faster rate of accumulation of deleterious mutations. We use Monte-Carlo simulations to quantify the interaction of niche breadth with accumulating deleterious mutations. These two mechanisms may act synergistically to prevent the extinction of the sexual population, given: (1) sufficient genetic variation, and consequently niche breadth, in the sexual population; (2) a relatively slow rate of accumulation of genetic diversity in the clonal population; (3) synergistic epistasis in the accumulation of deleterious mutations.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

Sexual reproduction suffers a two-fold fitness cost in comparison with asexual reproduction because of the investment of half the reproductive output in male gametes that do not themselves produce offspring (Maynard Smith, 1978). This ‘cost of males’ makes anisogamous sexual populations vulnerable to invasion by asexual mutants arising from within the sexual population. The two-fold advantage held by asexual clones gives them the potential to invade very rapidly. For example, Lively (1996) estimates that a population of 1 million sexual individuals would be replaced in <50 generations by an asexual clone. In attempting to understand why organisms reproduce sexually it is useful to ask ‘why are sexual populations not replaced by more efficient asexual competitors?’ (Maynard Smith, 1978).

The frozen niche variation hypothesis (Vrijenhoek, 1979) predicts that clones arising from within a sexual population will represent a sample of the genetic variation within the sexual population. In consequence the resources exploited by a clone may represent a fraction of the total niche exploited by the original sexual population. The outcome of an asexual invasion in an ecological timescale under the frozen niche variation hypothesis depends on the inter-specific interactions between sexual and asexual competitors (Doncaster et al., 2000). This hypothesis is distinct from Bell's model of sib-competition (Bell, 1982) in which the dynamic influences on coexistence are intra-specific, although the two models have been confused (e.g. West & Peters, 2000; see response in Pound et al., 2002).

Several studies have developed population dynamic models consistent with the frozen niche variation hypothesis, which explore the threshold differences in niche overlap or breadth that permit coexistence (Case & Taper, 1986; Doncaster et al., 2000, 2003; Kerszberg, 2000; Pound et al., 2002). However, these models alone fail to address two concerns. First, the period of coexistence may last for only a finite period due to the accumulation of clonal diversity (Weeks, 1993); secondly, sympatric sexual and asexual morphs are rarely observed in nature (Bell, 1982).

This paper examines the behaviour of the frozen niche variation model over time by considering the outcome of an asexual invasion over an evolutionary timescale. We will explicitly model the evolution of clonal diversity through mutations to loci that control the ecological niche of asexual individuals. In particular, we will test the hypothesis that differences between the niches of competing sexual and asexual populations provide a time-window for the expression of longer term benefits of sexual recombination. One such theoretical benefit of sexual reproduction is the difference between the rates at which sexual and asexual populations accumulate deleterious mutations (Kondrashov, 1993).

During reproduction, mutations may occur at a number of sites that impact upon individual fitness. These mutations are typically deleterious to the fitness of their offspring. Sexual recombination produces greater variation in the fitnesses of sexual offspring than in the fitnesses of offspring produced clonally. Mutations are thereby removed from a sexual population by selection with greater efficiency than from an asexual population. Mutations will accumulate deterministically above a threshold rate of mutation per genome per generation (U). Although the rate at which mutations arise in natural populations is currently unclear, the deterministic model of mutation accumulation predicts that sexual reproduction is protected from invasion by asexual clones if its rate of mutation per genome per generation exceeds unity (U > 1), under conditions of epistasis between mutations or truncation selection (Kondrashov, 1988).

Deleterious mutations with a slight effect on the fitness of an individual may also accumulate stochastically in large populations where random drift is unimportant, because each has such a slight effect on fitness that the mutation is not selected out of the population (Kondrashov, 1995). The cumulative effect of very slightly deleterious mutations (VSDMs) on individual fitness, the mutational load, may become significant over evolutionary timescales. Asexual populations accumulate VSDMs at a significantly faster rate than sexual populations, from which VSDMs are purged by recombination. The equilibrium mutational loads of clonal populations are therefore higher than sexual populations. Over evolutionary timescales and all else being equal, the average fitness of a sexual population will be significantly higher than that of an asexual population (Peck et al., 1997). However, this long-term advantage to a sexual population affords it little immediate protection from an asexual invasion (Kondrashov, 1995).

Peck et al. (1999) showed how the speed of an asexual invasion can be slowed by migration through a sexual metapopulation, thereby increasing the period of time over which VSDMs may accumulate in the asexual population. This significantly reduces the probability of a successful asexual invasion. The time-window effect observed here is relevant to the invasion of asexual mutants under the frozen niche variation model. However, unlike Peck et al.'s (1999) metapopulation model, the frozen niche variation hypothesis is an effect that derives directly from differences between the behaviours of sexual and asexual morphs (Vrijenhoek, 1979).

A pluralistic explanation for the maintenance of sexual reproduction has found favour in a recent review of the topic (West et al., 1999). However, such approaches in the past have typically focused on a synergy between mutation accumulation and the frequency-dependent selection of host genotypes driven by parasites (Howard & Lively, 1994, 1998). In this paper, the accumulation of deleterious mutations will be incorporated into stochastic simulations of the frozen niche variation hypothesis. These simulations explicitly model the range of niches exploited by a sexual population, and the magnitude of competition between phenotypes.

We simulate the accumulation of clonal diversity through mutations to genes affecting the ecological niche, and the decline in individual fitness because of the accumulation of deleterious mutations. The potential synergy between these evolutionary and ecological mechanisms has not previously been investigated. We use Monte-Carlo simulations of asexual invasions of a sexual population over a range of parameter values for mutation rate, the number and width of niches and epistatic interactions between mutations, to determine the conditions under which a sexual population excludes an asexual competitor.

Methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

We first model niche-specific fitness and the evolution of clonal diversity; we then build into this model the accumulation of VSDMs. The algorithm used has two components. The first component models the genetic diversity that determines niche breadth (the first Section below). The second component uses Peck et al.'s (1997) model of VSDMs to simulate the accumulation of deleterious mutations within populations (the second Section below).

Estimating niche-specific fitness

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

Consider a system in which a polymorphic sexual population exploits a range of ecological niches, where each niche has a carrying capacity of N individuals. The total resource niche of the sexual population includes all of the niches exploited by all of the sexual individuals. Niches are defined by a number of ecological characteristics that influence how efficiently they are utilized by sexual individuals within the population.

An individual's ability to exploit different niches is the product of several genetically controlled phenotypic traits, for example, sensitivity to light or moisture. This is consistent with Hutchinson's description of the ecological range of a species as defined by a number of ecological variables (Schoener, 1986). By describing the genes that control an individual's ecological range we are defining its niche phenotype. The niche phenotype is controlled by n loci, each with two alleles. Every combination of these alleles encodes a unique phenotype, so that within the system there are P = 2n possible niche phenotypes. For simplicity in these simulations every phenotype has a corresponding optimal niche.

The relationship between niche phenotypes may be visualized as a hypercube. For example, Seger & Hamilton (1988) used hypercubes to describe the relationship between loci determining resistance to parasites. Each phenotype has n nearest neighbours and is separated from its opposite by n unique mutations. Figure 1 shows an example of a 3-loci system.

image

Figure 1. The relationship between the eight niche phenotypes for a 3-loci system.

Download figure to PowerPoint

Each phenotype exploits the niche to which it is best adapted with a niche-specific fitness, wP, of unity. The fitness of each phenotype at any other niche is determined by the genetic distance, d(x,y), separating the phenotype, x, from the optimum phenotype for that niche, y:

  • image(1)

where, d(x,y) is the number of unmatched loci between phenotypes x and y. In Fig. 1, for example, when x = [0,1,1] and y = [1,0,1] then d(x,y) = 2. The parameter L controls the phenotypic niche breadth by modifying the fitness of phenotypes in nonoptimal niches. When L = 1 the niche-specific fitness of a phenotype decreases in direct proportion to its distance from the optimal phenotype, but when L = 5 the fitness of a phenotype declines sharply in nonoptimal niches (Fig. 2). The fitness of a phenotype in a niche for which it has no matching alleles is always zero.

image

Figure 2. Relationship between niche-specific fitness, wP, of a phenotype, x, and its genetic distance from the optimum phenotype for a niche, y [eqn  (1)]. This example shows a 3-loci model with (a) a broad niche (L = 1), and (b) a narrow niche (L = 5).

Download figure to PowerPoint

 Each generation all adults reproduce, sexual individuals recombining with a randomly chosen partner. The loci controlling the niche phenotype segregate independently. During reproduction all gametes are prone to undergo mutation. The loci controlling niche phenotype mutate with a probability μP per locus, per generation. Sexual recombination and the mutation of loci controlling niche exploitation will produce variation between the niches that offspring are most able to exploit.

An asexually reproducing population will accumulate clonal diversity through mutation of the niche exploitation loci. Clonal diversity may also accumulate through repeated mutations to asexuality from the sexual population (Weeks, 1993). However, for simplicity, only the mutation of the niche exploitation loci is modelled here, allowing us to estimate the period of coexistence following an asexual mutation.

Previous models (Doncaster et al., 2000; Pound et al., 2002) explicitly examined the population dynamics of competing species using Lotka-Volterra equations for the competition between two reproductively isolated populations, in which the per capita intrinsic rate of growth of each population was partitioned into birth and death rates. Under this approach, differences between the realized birth and death rates of competing sexual and asexual populations are expressed in the equilibrium population sizes, and traits are selected under a scheme of ‘hard’ selection. In anticipation of the high mutational loads common to theoretical models of the accumulation of deleterious mutation (Peck et al., 1997), here we employ a scheme of ‘soft’ truncation selection to model competition for places within a niche. Offspring are selected to fill N available places in each niche in proportion to their niche-specific fitness, wP. Individuals thus may occupy nonoptimal niches, but not niches for which they have none of the required phenotypic traits (where wP = 0).

The asexual invasion is initiated by a single mutation to obligate asexual reproduction within the sexual population. This mutation involves the suspension of the production of male gametes and the spontaneous development of female gametes. The mutation from obligate sexual reproduction to clonal reproduction may be considered an extreme example of the evolution of investment in sexual reproduction by a species that employs a mixed strategy of sexual and asexual reproduction.

The sexual population considered here is hermaphroditic, such that each individual devotes half its resources to the production of male gametes. The two-fold cost of sexual reproduction applies equally to sexual populations with separate sexes (Maynard Smith, 1978). Its consequence to reproductive output is that sexual individuals have half the resources available to asexual individuals for production of females’ gametes. Therefore, during reproduction sexual individuals produce half the number of offspring of asexual individuals. The offspring of an asexual individual will be represented twice as often in the pool of offspring available for selection; this confers a two-fold fitness advantage to asexual reproduction, all else being equal.

Figure 3 compares the average time taken for asexual mutants to invade a sexual population for different parameters. The default parameter values (Fig. 3a) describe a species in which phenotypes are highly niche-specific (L = 5), the rate of mutation of niche exploitation loci is initially set equal to an intermediate rate of deleterious mutations (μP = μ = 0.001; see Peck et al., 1997), and the resource niches each support an intermediate number of individuals (N = 50). Sexual populations were simulated for systems in which the number of niche phenotypes, P, varied between 1 and 16.

image

Figure 3. In the absence of accumulating deleterious mutations, the number of generations taken for five asexual mutants to exclude the original sexual population (mean values and 95% confidence limits from 100 simulations at each point). Parameter sets: (a) N = 50, L = 5 and μP = 0.001; (b) N = 10, L = 5 and μP = 0.001; (c) N = 100, L = 5 and μP = 0.001; (d) N = 50, L = 5 and μP = 0.01; (e) N = 50, L = 1 and μP = 0.001.

Download figure to PowerPoint

In Fig. 3, we see that the sexual populations with greater genetic diversity (further to the right), and therefore exploiting more niches, typically take longer to be excluded by the increasingly diverse clonal population. However, the number of mutations of niche exploitation loci per generation within the asexual population increases in proportion to its total population size. The total number of individuals supported by a system is equal to the number of niches, P, multiplied by the carrying capacity within each niche, N. Thus, the ability of the asexual population to generate new phenotypes also increases with the number of phenotypes in the system. This is seen in parameter set (Fig. 3b) where the small niche carrying capacity slows the generation of mutations of niche exploitation loci in the asexual population compared with Fig. 3a, and the speed of an asexual invasion increases as P increases above 4. The speed of an asexual invasion also increases with a higher niche carrying capacity, N (set Fig. 3c compared with Fig. 3a), a higher rate of mutation of niche exploitation loci, μP (set Fig. 3d), and a higher phenotypic niche breadth, L (set Fig. 3e).

Accumulation of very slightly deleterious mutations

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

The accumulation of VSDMs is based on a scheme designed by Peck et al. (1997). Each individual has 50 loci on each of 10 diploid chromosomes, giving a total of 1000 genes that may undergo deleterious mutation. To model the influence of VSDMs each gene is represented by a floating-point number, the optimum state of which is zero. The cumulative impact of VSDMs upon the fitness of the gene is represented by the absolute value of this number. For this analysis, the effect of VSDMs upon the loci controlling the niche phenotype is not considered.

At each generation a gene has a probability μ of undergoing a mutation event. The probability of mutation per locus per generation used initially is μ = 0.001, to give a genomic mutation rate per generation of U = 1.0. This value of U represents a theoretical threshold for deterministic accumulation of mutations in the sexual population (Kondrashov, 1988), which has been applied to an earlier study of the effect of VSDMs upon asexual invasion (Peck et al., 1997). We will also consider the effects of other possible values of μ.

A mutation alters the fitness of the gene by a randomly generated positive or negative number, m. Peck et al. (1997) determined the distribution of impact of each mutation from a reflected gamma distribution:

  • image(2)

where, Γ(0.5) is a gamma function with parameter 0.5, and a value of α = 24.253 scales the distribution to ensure that mutations to near-perfect alleles will, on average, decrease fitness by about 2%. This distribution produces a range of mutational effects, including a large proportion of VSDMs. For the simulations presented here, however, we use a Gaussian distribution, which is less computationally expensive than the reflected gamma distribution. The Gaussian distribution is given by:

  • image(3)

where, the SD, σ = 0.025. This value scales the distribution to give an average absolute mutational effect, |m| of 0.02, which corresponds to the average mutational effect measured empirically (Peck et al., 1997).

The mutational load of an individual, which represents its distance from optimum fitness (Peck et al., 1997), is a function of the total impact of VSDMs on the fitness of all genes, Z, raised to the power of E. The value of Z for individual k is given by:

  • image(4)

where, |xi,j,k| is the distance of the ith locus on chromosome j from its optimal fitness. Parameter E controls the level of epistatic interactions between the loci, and when E = 1 the model is multiplicative. When E > 1 there is synergistic epistasis between deleterious mutations, whereby individuals suffer a greater cost to their fitness resulting from the cumulative effects of all mutations than is given by the sum of the individual mutational effects. The fitness, wk, of individual k is given by:

  • image(5)

We now incorporate this scheme into the niche model described in the previous Section. The chromosomes segregate individually during sexual recombination. Adjacent loci on the same chromosome recombine with a frequency r = 0.016. When offspring are selected to fill a niche, the individual's fitness, wk, is multiplied by the niche-specific fitness of its phenotype, wP, to produce an individual niche-specific fitness:

  • image(6)

Individuals are selected to fill places within a niche in proportion to this value.

When the model is multiplicative, given by E = 1 in eqn  (5), a population at equilibrium will have a massive mutational load as a result of the stochastic accumulation of VSDMs. Whether such populations can persist is the subject of some debate (Peck et al., 1997). When there is synergistic epistasis between deleterious mutations, given by E > 1, each additional mutation is subject to a greater strength of selection. Under these conditions the mutational load of a population at equilibrium will be several orders of magnitude smaller than under the multiplicative model. Such synergistic epistasis between deleterious mutations is thought to be significant in the interactions between VSDMs, albeit with more complex interactions than modelled here (Kondrashov, 1995; Rice, 1998). Most of the conditions studied here involve synergistic epistasis between deleterious mutations in which E = 8.

Sexual recombination produces offspring with a range of fitnesses, allowing selection to purge the population of deleterious mutations. Consequently deleterious mutations accumulate faster in an asexual population than a sexual population. This means that over time asexual mutants invading a sexual population suffer a drop in fitness relative to the sexual population. In small asexual populations, the rate of accumulation of VSDMs may be increased by Muller's ratchet, which describes the fixation of deleterious mutations in an asexual population because of random drift (Muller, 1964; Kondrashov, 1995).

The success of an asexual invasion depends on the ability of the asexual population to exclude its sexual ancestors before it accumulates a mutation load that counterbalances the two-fold ‘cost of males’ which has constituted its advantage over the sexual population. The speed of the asexual invasion is therefore crucial to its success.

The ability of asexual clones to invade a genetically diverse sexual population was tested across broad areas of parameter space by generating asexual mutants from within simulated sexual populations. For efficiency, the sexual populations were precomputed for 100 000 generations to ensure they reached an equilibrium mutational load starting from an ideal fitness of zero. For each point in parameter space, five precomputed sexual populations were produced and allowed to mutate for a further 500 generations before each was used to generate 20 asexual invasions. A mean response value with 95% confidence limits was then computed from these 100 simulations. Each invasion was initiated by switching the gene that determines the method of reproduction in five sexual individuals of the same phenotype to produce five asexual clones. More than one asexual clone was used in order to prevent stochastic asexual extinctions during the start of an invasion.

Asexual invasion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

Figure 4 shows the equilibrium individual fitness within sexual populations prior to an asexual invasion. The mutational load of a sexual population decreases with increasing population size, and so fitnesses are higher for parameter sets with higher numbers of phenotypes, and thus larger total population sizes (further to the right of the graph). Fitness also increases significantly between populations with higher niche carrying capacities, from N = 10 to 50 and 100 (Fig. 4a–c respectively). The mutational load is little influenced by the rate of mutation of niche exploitation loci, μP (Fig. 4d) and niche breadth, L (Fig. 4e).

image

Figure 4. Equilibrium fitness of sexual populations following 100 000 generations (mean values and 95% confidence limits from five simulations at each point). Parameter sets as for Figure 3. All runs use μ = 0.001 and E = 8.

Download figure to PowerPoint

Figure 5 shows the probability of success for an asexual invasion into a sexual population. In systems with a single niche phenotype (P = 1), there is no difference between resources utilized by the sexual population and the asexual invaders. Successful invasions occur with probability <1 because of the fast accumulation of deleterious mutations in the initially small clonal population. The probability of a successful asexual invasion drops markedly in sexual populations with more niche phenotypes (Fig. 5, further to the right-hand side of the graph). This is because a mutation at a locus controlling niche exploitation must occur within the asexual population before the asexual clones can spread into a new niche.

image

Figure 5. Effect of niche carrying capacity, N, and niche breadth, L, on the probability of successful asexual invasion into a sexual population (averaged from 100 simulations at each point). All runs used μP = μ = 0.001 and E = 8.

Download figure to PowerPoint

Under the conditions shown in Fig. 5, of narrow niche breadth and an intermediate rate of mutation of niche exploitation loci (L = 5, μP = 0.001), the probability of a successful asexual invasion across four or more niches is typically close to zero. For these parameters the number of individuals supported by each resource niche, N, has little impact on the success of an asexual invasion in systems with more than one niche phenotype. Where L = 1, the niche of each phenotype is wider, and the interaction between neighbouring phenotypes is greater. Comparison of Fig. 5a and d shows that the width of the phenotypic niche has a minor influence on the probability of an asexual invasion, with the wider niche favouring the asexual population.

Figure 6a shows how the equilibrium fitness of the sexual population increases with total population capacity, where Ntot = N·P. However, the probability of a successful asexual invasion is largely independent of the total population capacity (Fig. 6b). This means that the outcome of an asexual invasion is little influenced either by the variation in equilibrium fitness of the sexual population with Ntot, or by the size of the initial asexual population relative to the total size of the sexual population.

image

Figure 6. Effect of total population capacities, Ntot = N·P, on (a) equilibrium fitness of a sexual populationinline image, and (b) the outcome of an asexual invasion (averaged from 100 simulations at each point). Parameter values for P increasing from 1 to 16: (i) Ntot = 50, 100, 200, 400, 800; (ii) Ntot = 200, 200, 200, 200, 192. All runs use L = 5, μP = μ = 0.001 and E = 8.

Download figure to PowerPoint

Figure 7 further explores the outcome of asexual invasions across parameter space for mutation rates lower than μ = 0.001, and simultaneously for a range of values for the rate of mutation of niche exploitation loci, μP and magnitude of synergistic epistasis, E. The mutation rate μ has a stronger influence on the probability of a successful asexual invasion than either μP or E. The ability of the sexual population to survive asexual invasion is typically zero at the lowest mutation rate simulated, μ = 0.0001. Additional simulations have shown that the asexual population is largely unable to invade the sexual population under the highest mutation rate, μ = 0.01, except for conditions of no synergistic epistasis (E = 1) and a single phenotypic niche (P = 1). Between these values of μ, the rate of mutation of niche exploitation loci influences the outcome of an asexual invasion. Lower μP reduces the speed with which the total niche of asexual population expands as the asexual population mutates although the hypercube (e.g. Fig. 1), reducing the probability of asexual invasion.

image

Figure 7. Effects of the rate of mutation of fitness affecting loci, μ, the rate of mutation of niche exploitation loci, μP and synergistic epistasis, E, upon the outcome of an asexual invasion. The probability of successful asexual invasion at each point is averaged from 100 simulations. All runs used N = 50 and L = 5.

Download figure to PowerPoint

The effect of synergistic epistasis on the success of an asexual invasion is examined in Fig. 8a. It shows a substantially higher equilibrium fitness of sexual populations under synergistic epistasis (E = 8) than under multiplicative interactions between mutations (E = 1). In effect, the impact on fitness of each additional mutation is magnified by any amount of synergistic epistasis, producing a stronger selection against deleterious mutations. Under synergistic epistasis the greater impact of each deleterious mutation means that the fitness of asexual clones arising in the sexual population drops more rapidly than under the multiplicative model (Fig. 8b). Under the multiplicative model the greater length of time taken for the fitness of an asexual population to halve means that there is a larger time-window available to the asexual population to invade the sexual population, increasing the probability of a successful asexual invasion. Interestingly, Fig. 7 shows that intermediate values of E = 2 and 4 yield similar probabilities of a successful asexual invasion to E = 8.

image

Figure 8. Effect of synergistic epistasis on (a) equilibrium fitness of sexual populations following 100 000 generations (averaged from five simulations at each point), and (b) time taken to halve fitness of asexual invaders (averaged from 100 simulations at each point). All runs used N = 50, L = 5 and μP = μ = 0.001.

Download figure to PowerPoint

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

These results demonstrate that a synergy between the frozen niche variation hypothesis (Vrijenhoek, 1979) and the accumulation of deleterious mutations (Kondrashov, 1988) may provide a stronger explanation for the ubiquitous presence of sex throughout nature than either mechanism alone. Three key factors determine the outcome of an asexual invasion in the systems that we have modelled: (1) the genetic variation in the sexual population, controlling niche breadth; (2) the rate at which clonal diversity accumulates in the asexual population; (3) the rate at which the fitness of the asexual population declines as a result of the accumulation of deleterious mutations. All three of these factors are system-dependent and they vary widely in nature.

Niche phenotype

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

The coexistence of sexual and asexual morphs under the frozen niche variation hypothesis depends crucially on the amount of genetic variation present in the sexual population, controlling the resource exploitation by individual phenotypes. We have modelled genetic variation in terms of the number of phenotypes in a sexual population exploiting P resource niches, and the competitive interaction between these phenotypes, represented by L. Our generic model has allowed us to examine a range of examples, from a sexual population consisting of a single general purpose phenotype (P = 1) to one in which numerous sexual phenotypes each exploit a highly specific resource niche (P = 16, L = 5).

Our definition of resource niches in terms of phenotypic traits is consistent with the definition of an ecological niche described by multiple dimensions (Schoener, 1986). Our phenomenological model has each phenotypic trait controlled by a single locus, which provides us with a simple way of representing genetic control of the phenotypic traits that determine the ecological niche of the individual. Although this model is clearly a simplification of the genetic control of multiple phenotypic traits, it provides a useful first approximation to niche dimensions that avoids the complexity that arises with more detailed models. The representation of niche dimensions is the key to understanding interactions between the frozen niche variation hypothesis and the unrelated mechanism of the accumulation of deleterious mutations.

An alternative approach is to consider the behaviour of a population exploiting resources along a single ecological gradient (Case & Taper, 1986; Doebeli, 1996). Doebeli (1996) modelled a system in which the phenotype of an individual was controlled by numerous loci encoding a single phenotypic trait, the phenotype expressed being the sum of all alleles. This allowed the analysis of resource partitioning between populations competing along a single ecological gradient. Doebeli's (1996) model differs from the one used here in a number of ways. For example, a sexual population in the absence of any selection will recombine to produce a normal distribution of phenotypes. In contrast, the model used here is nondirectional and all phenotypes can potentially exploit an equal quantity of resources. This model has allowed us to quantify the reality of ecological niches that are defined by multiple dimensions.

Genetic variation in the sexual population for the resource exploited along a single dimension, given by P = 2, had a significant impact upon the success of an asexual invasion (Fig. 5). It must be remembered that when P = 2, individuals of the two opposing phenotypes have no impact upon one another. The potential for coexistence between species is determined by some measure of limiting similarity (Schoener, 1986), and all simulations here investigated parameter sets in which phenotypes exploited relatively narrow resource niches, given by L ≥ 1. Clearly the impediment to an asexual invasion is reduced under conditions of broader phenotypic niches, simulated by values of L < 1.

The amount of genetic variation for phenotypic resource use within a sexual population will depend on the complexity of the biotic and abiotic environments, and competitive interactions with other species. Interspecific interactions are certainly fundamental in determining the niche exploited by a population. Van Valen's (1965)‘niche-variation hypothesis’ predicted that the between-phenotype variance for resource utilization should expand in the absence of interspecific competition.

Clonal diversity

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

The rate at which clonal diversity accumulates in the asexual population was controlled in our model by the rate of mutation of niche exploitation loci per locus per generation, μP. The default value of μP used here was initially equal to the rate of deleterious mutations per locus, per generation, μ (Fig. 5). However, these values do not need to be associated, and μP has an independent influence upon the speed (cf. Fig. 3a and d) and success (Fig. 7) of an asexual invasion. The rate at which mutations occur in the loci controlling the niche phenotype will depend on the system concerned, but clearly a lower μP relaxes the conditions for coexistence.

The accumulation of clonal diversity through mutation of niche exploitation loci is analogous to the production of clonal diversity by repeated mutations to asexuality from the sexual population, modelled by Weeks (1993). There are, however, important differences between the two phenomena.

First, the occurrence within the asexual population of mutations to loci controlling niche exploitation is proportional to the size of the asexual population, whereas the rate at which new asexual clones are generated is proportional to the size of the sexual population. It is unclear what effect this has on the ability of clones to invade a sexual population. When clones arise repeatedly from the sexual population, new clones will arise more quickly when the sexual population is large. This raises the potential for phenotypic ‘refugees’ of small sexual populations following an asexual invasion. By contrast, when clonal diversity is generated by mutation of niche exploitation loci within the asexual population, an asexual invasion gains momentum by virtue of its increasing population size.

Secondly, clones arising by mutation from the sexual population inherit the mutational load of their sexual parent. When clones arise repeatedly, the sexual population will face competition from clonal strains with different fitnesses, which is likely to benefit the asexual mode. In nature, clonal diversity may accumulate simultaneously through both mutation of niche exploitation loci and repeated clonal invasions, depending on the system.

Mutational load

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

The concept of mutational load presents a number of theoretical difficulties, particularly because it represents the distance from an immeasurable ideal: the optimum fitness for an organism (Rice, 1998). The high mutational loads, especially under the multiplicative model, appear incompatible with population growth under hard selection. Whilst the transition, through mutation accumulation, of a population from an ideal optimum to the equilibrium of condition of high mutational load is a theoretical abstraction, it nevertheless provides a useful tool for studying the behaviour of asexual populations arising under conditions of a mutational equilibrium in the sexual population.

The rate at which the fitness of asexual clones declines following an invasion depends on the rate μ at which mutations arise, the distribution of mutational effects, the interaction between mutations (characterized by E in our model), and the population size. The mutation rate clearly has a strong influence on the outcome of an asexual invasion (Fig. 7). At very low rates of mutation the accumulation of deleterious mutations in clonal lineages is too slow to prevent the extinction of the sexual population, even when the invasion is checked by differences in the niches of the sexual and clonal populations (i.e. P > 1). Conversely larger mutation rates increase the rate at which the fitness of the asexual population declines, thereby increasing the ability for the sexual population to resist invasion.

The parameter space explored in Fig. 7 shows that differences between ecological niches of the sexual and asexual populations (where P > 1) keeps the sexual population viable below μ = 0.001. The mutation rate per locus μ = 0.001 corresponds to a mutation rate per genome per generation of U = 1. This genomic mutation rate also represents a theoretical threshold above which sexual reproduction is viable under the deterministic model of mutation accumulation (Kondrashov, 1988). The value of the genomic mutation rate, U, of different species remains the subject of a great deal of research, with estimated values for higher eukaryotes ranging 0.1  < U < 100 (Drake et al., 1998; Lynch et al., 1999). Although hominids have amongst the highest known rates of deleterious mutations (Eyre-Walker & Keightley, 1999), they are largely protected from asexual invasion by developmental constraints. However, the mutation rate of Drosophila, a more suitable model organism, has been re-estimated considerably lower than previous measurements, at U << 1 (Keightley & Eyre-Walker, 2000; cf. Kondrashov, 2001). These estimates suggest that for many taxa the rate of production of deleterious mutations may lie below the theoretical threshold at which sexual populations are viable under the deterministic model of mutation accumulation (Lynch et al., 1999). Our results demonstrate, however, that differences between the niches of sexual and asexual populations expected by the frozen niche variation hypothesis can potentially lower the threshold mutation rate under which the sexual population is capable of out-competing an invading clonal population (Fig. 7).

The genomic rate cannot be considered in isolation from the range and distribution of mutational effects on the fitness of the individual (Keightly, 1994). Peck et al. (1997) suggest that previous attempts to measure U may well have underestimated the rate of mutations that are very slightly deleterious to the fitness of an individual. This is important as the mechanism for the accumulation of deleterious mutations modelled here is a result of the stochastic accumulation of VSDMs in sexual and asexual populations. Additional simulations showed qualitatively similar results to those presented in Fig. 5a when a reflected gamma distribution with an average absolute mutational effect, |m|, of 0.02 was used in place of the Gaussian distribution. The Gaussian distribution used here (or indeed the reflected gamma distribution) are approximations of the actual distribution of mutational effects in nature that may in fact be the combination of a number of distributions (Keightly, 1994).

Muller's ratchet may also influence the rate at which deleterious mutations accumulate in small populations (Kondrashov, 1995). The fixation of deleterious mutations by random genetic drift may be relevant to the fitness of an asexual population during the initial stages of an invasion, or to an asexual population that is restricted by a limited resource niche. However, for the parameters studied here the number of individuals supported by a niche had little influence upon the outcome of an asexual invasion (Fig. 6b).The model of Peck et al. (1997) differs from alternative stochastic models of mutational accumulation by Lande (1994) and Schultz & Lynch (1997) in the way that it represents the effects of individual mutations. Under the scheme used by Peck et al. (1997), mutation events are either beneficial or deleterious depending upon the fitness state of the gene that they influence. In contrast, the schemes of Lande (1994) and of Schultz & Lynch (1997) have fixed probabilities for the occurrence of deleterious and beneficial mutations, with the result that the impact of the mutation is not conditional on the fitness state of the gene. This assumption produces dynamics that depend strongly on population size, with small populations being particularly prone to extinction.

Synergistic interactions between deleterious mutations greatly increase the rate of decline in fitness of the asexual population in comparison with the multiplicative model (Fig. 8b). This is because the impact of each additional mutation is greater, which results in a faster increase in the mutational load of the asexual population (Peck et al., 1997). However, differences between the affect upon the probability a successful asexual invasion of synergistic interactions of different magnitude was slight (cf. E > 1 in Fig. 7).

Pluralism and the maintenance of sexual reproduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References

Our analysis of the interactions between the two independent mechanisms of frozen niche variation and accumulating deleterious mutations has suggested that a synergy between them may relax the conditions under which sexual reproduction survives asexual invasion. However, there are concerns in adopting a pluralistic approach.

In response to the suggestion of West et al. (1999) that sexual reproduction was maintained by the interaction of more than one mechanism, Kondrashov (1999, p. 1031) replied: ‘…such a beautiful phenomenon as sex deserves a nice, simple explanation and messy interactions of very different processes would spoil the story’. Whilst Occam's razor compels us to accept the most parsimonious explanation, it would be short-sighted to reject a pluralistic explanation out of hand. Indeed the conditions that Kondrashov (1999) requires before a pluralistic explanation is acceptable are that its individual components are both important and individually insufficient.

The accumulation of deleterious mutations is clearly an important factor in the maintenance of sexual reproduction. However, the rate at which deleterious mutations accumulate deterministically, and the advantage conferred to sexual reproduction at mutation-selection balance, depends on the genomic mutation rate, U. This value, and therefore the ability of sexual reproduction to resist an asexual invasion, has not been widely resolved (Kondrashov, 1999). The impact of VSDMs, which may accumulate stochastically, is of limited relevance in the short-term (Kondrashov, 1995; Peck et al., 1997, 1999).

The frozen niche variation hypothesis has been implicated in empirical studies of coexistence between sexual and asexual morphs (Case, 1990; Christensen et al., 1992; Barata et al., 1996; Fox et al., 1996; Vrijenhoek & Pfeiler, 1997; Negovetic et al., 2001). Whilst frozen niche variation alone may be insufficient to withstand repeated invasion by asexual mutants (Weeks, 1993) it has the potential to significantly influence the course of an asexual invasion (Pound et al., 2002).

The frozen niche variation hypothesis has been overlooked by theoreticians examining pluralistic hypotheses for the maintenance of sexual reproduction (Kover & Szathmary, 1999). However, the interaction between the frozen niche variation hypothesis and the accumulation of deleterious mutations is intuitively appealing, however, because it gives an ecological context to the evolutionary problem of why sex prevails in nature. We have demonstrated that there is great potential for these two mechanisms to interact synergistically to prevent the extinction of the sexual population competing with a diverse clonal population, and without the addition of further assumptions. The frozen niche variation hypothesis should be considered amongst the potential explanations for the geographical distribution of sexual reproduction (Bell, 1982).

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Estimating niche-specific fitness
  6. Accumulation of very slightly deleterious mutations
  7. Results
  8. Asexual invasion
  9. Discussion
  10. Niche phenotype
  11. Clonal diversity
  12. Mutational load
  13. Pluralism and the maintenance of sexual reproduction
  14. Acknowledgments
  15. References