### Abstract

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

**Abstract** The genetic variation in a partially asexual organism is investigated by two models suited for different time scales. Only selectively neutral variation is considered. Model 1 shows, by the use of a coalescence argument, that three sexually derived individuals per generation are sufficient to give a population the same pattern of allelic variation as found in fully sexually reproducing organisms. With less than one sexual event every third generation, the characteristic pattern expected for asexual organisms appear, with strong allelic divergence between the gene copies in individuals. At intermediary levels of sexuality, a complex situation reigns. The pair-wise allelic divergence under partial sexuality exceeds, however, always the corresponding value under full sexuality. These results apply to large populations with stable reproductive systems. In a more general framework, Model 2 shows that a small number of sexual individuals per generation is sufficient to make an apparently asexual population highly genotypically variable. The time scale in terms of generations needed to produce this effect is given by the population size and the inverse of the rate of sexuality.

### Introduction

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

Many higher organisms have both sexual and asexual means of letting themselves be represented in future generations. In plants partial asexuality is normally due to a mixture of sexual reproduction by seed and asexual reproduction by specialized structures such as runners or bulbils, whereas in animals partial asexuality often follows from the failure of cyclical parthenogens to go through the sexual stage successfully.

The present article considers the pattern of selectively neutral genetic variation expected in organisms with a stable mixture of sexual and asexual reproduction. Asexuality has classically been regarded as a factor that reduces genetic variation. As a general notion, this is, of course, not correct. Asexual species often harbour a wealth of variation (Ellstrand & Roose, 1987; Hebert, 1987; Suomalainen *et al*., 1987; Asker & Jerling, 1992) coming from new mutations as well as remnant sexuality and/or multiple origins. But *how* genetically variable do we expect a predominantly asexual organism to be?

From models of infinitely large populations is known (Marshal & Weir, 1979), that asexuality as such does not affect the equilibrium genotype frequencies of neutral alleles in organisms practicing at least some outbreeding sexuality. The only role asexuality plays in such reproductively mixed conditions is to slow down the rates at which the multilocus equilibrium values are attained (Marshal & Weir, 1979). Asexuality differs in this respect from, say, inbreeding, in that it does not change the genetic structure of a population in any specific, unidirectional way.

Although this insight is valuable, one can ask how relevant it is for actual populations of limited size in which genetic drift cannot be ignored. Two immediate questions arise for such populations. The first concerns the degree of *allelic* variation expected at loci at which neutral mutations occur. Asexuality affects this variation in two different ways. On the one hand, asexuality implies that sampling drift acts at the level of individuals as well as on the level of genes, which will lead to an increased homogenization at the population level. On the other hand, asexuallity tends to ‘lock up’ gene combinations in fixed pairs that are inherited together, which will tend to increase the divergence between gene copies. By use of a coalescence argument in Model 1 below, I show how these processes interact and describe the expected pattern of pairwise allelic divergence in a population with a fixed degree of asexuality. The results are obtained under the assumption that the investigated population is large and that it has had a constant size for a long time.

The second question concerns *genotypic* variation. With the advent of methods for multilocus genotyping (isozymes, RAPD, AFLP and other techniques), the genotypic, ‘clonal’, variation in populations of organisms with different degrees of asexuality has become easy to investigate empirically. It has then been found that apparently asexual organisms often harbour considerable genotypic variability. To assess the basis for this variation – is it because of low levels of sexuality or to factors such as somatic mutations, balancing selection, or perpetual re-creation of the asexual form? – the expected level of neutral genotypic variation in a limited population with partial asexuality must be known for comparison. This question is investigated in Model 2, which is less restricted than Model 1 with respect to the underlying assumptions made. Numerical examples are used to describe the most important results, many of which are already directly or indirectly known.

It is assumed in both models that the organism investigated has its dominant, size-limited life-stage at the higher ploidy level (the interesting question of genetic variation in mosses, for example, thus not being considered) and that infinitely many gametes are produced during sexual reproduction. Unless otherwise indicated, sexual reproduction is assumed to occur through outbreeding. Generations are assumed to follow each other in discrete and separate steps.

Earlier theoretical analyses of fully or partially asexual organisms (see, e.g. Lokki, 1976a,b; Pamilo, 1987; Brookfield, 1992) have dealt primarily with the degree of variation expected for particular genetic markers. My aim is to generalize the type of situations considered and to clarify the interactions between the evolutionary processes involved. The approximations used in the first model apply to very long time scales in which an equilibrium in the population is established between mutation and drift, whereas the second model deals with shorter time scales for which reorganization by recombination of the variation already available is of greater importance. The results are of relevance for the question of what method to use for estimating past and current levels of sexuality/asexuality from population data. However, this is a complex topic (see various approaches in Marshall & Brown, 1974; Stoddard & Taylor, 1988; Maynard Smith & Smith, 1998; Mes, 1998; Maynard Smith, 1999; Ceplitis, 2000) that needs further developments. The aim of the present paper is limited to a description of the qualitative and quantitative interactions between the major parameters that govern neutral evolution in partially asexual populations.

### 2-gene coalescence and diversity

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

The aim of the analysis is to study the degree of divergence in the population between pairs of gene copies. To this end, the expected coalescence time for two copies drawn from the population at random is estimated (for a textbook introduction to coalescence analysis, see e.g. Gillespie, 1998). If an infinite-sites assumption regarding the structure of the gene is made (Gillespie, 1998; p. 31), the degree of sequence divergence between the copies is directly proportional to their coalescence time.

The gene copies can exist in two different states: either that of residing in different individuals, or of residing in the same individual as its two homologous gene copies at the locus in question. To estimate the mean time to coalescence in each of the two cases, the transition probabilities between these states when one moves back in time need to be considered. The mathematical procedures required for this are outlined in the appendix. There it can be seen that for fixed values of *σ* greater than *N*^{−1}, the mean time to coalescence for two gene copies is 2*N*, irrespective of whether the copies are taken from the same or from different individuals. Thus, as long as sexuality occurs reasonably often in the population, the time to coalescence – and with that the expected degree of mutational divergence – for gene copies under partial asexuality will be the same as for gene copies from fully random mating populations of equal size. (This result is obtained as the limiting value for an unboundedly increasing *N*).

Things become more interesting if one let the *number* of sexually derived individuals per generation be constant (in a population of unboundedly increasing size). In the appendix it is shown how the coalescence times then become determined by two geometric processes (again as an approximation when *N* goes towards infinity). The first process follows a geometric probability distribution with mean

- (1)

whereas the second process follows a geometric probability distribution with mean

- (2)

The mean of the first process goes from 2*N* for substantial *s*-values (sexuality being common) towards positive infinity when *s* decreases towards zero. The mean of the second process goes from 0 when sexuality is common towards *N* when *s* decreases.

Consider initially the mean time to coalescence for two gene copies taken from the same individual. Both when sex is common and when it is rare, the coalescence time is determined by the first process alone (see appendix). Thus, the mean time to coalescence goes from 2*N (*as indicated in the discussion above) towards positive infinity when *s* decreases towards zero (when there is no sex at all, the two gene copies contained in a diploid never coalesce). The second geometric process cannot be fully ignored, however, because it contributes to the value of the coalescence mean expected for intermediary values of *s* (see the example below in which *s* = 1).

The coalescence time for two gene copies taken from different individuals is determined by the first geometric process when *s* is large (again as expected). An interesting thing happens, however, when sexuality becomes increasingly rare. The relationship between the two gene copies can be described then as follows: their coalescence distribution is given with a probability of ½ by the first geometric process for which the mean goes towards infinity and with a probability of ½ by the second geometric process for which the mean goes towards *N*. Thus, when sexuality is very rare or is absent, the origin of two gene copies taken from different individuals in the population either can be traced back to the start of the present breeding system or they coalesce at an age of about half that for copies taken from a sexual population of equal size.

The advantage of performing a complete formal analysis becomes now evident, as it allows the transition from the state dominated by sexuality to that determined by full asexuality to be followed exactly. Figure 1 shows how the coalescence means change as functions of the number of sexual events in the population, *s*, as this value goes from 10 to 0.1 per generation [the *x*-axis in the figure corresponds to –log(*s*)]. It can be seen that within this range of sexuality the pattern of allelic variation in the population changes markedly. In the case of more than three sexually derived individuals per generation (*s* = 3 corresponds to *x* ≈ −0.5), both coalescence times considered are almost the same as for genes taken from a completely sexual population. In the case of less than one sexual event every third generation (*x* ≈ 0.5) the mean coalescence time for gene copies taken from the same individual is much larger than the mean coalescence time for gene copies taken from different individuals, both means being much larger than the corresponding mean expected under complete sexuality.

In the case of exactly one sexual event per generation (*s* = 1, *x* = 0), the coalescence mean for gene copies taken from the same individual is 4*N* and for copies taken from different individuals 3*N*, as compared with the value of 2*N* that holds for gene copies taken from a completely sexual population. For this value of *s* in the middle of the critical range of sexuality/asexuality, the two coalescence means are given by linear combinations of the means of the two underlying geometric processes (see appendix). Their means for *s* = 1 are (2 + √2)*N* (≈3.4*N*) and (2 − √2)*N* (≈ 0.6*N*), respectively.

Note that all coalescence processes described above have the properties characteristic of geometric distributions. Thus, the distributions are strictly and monotonically decreasing and have very large variances. The first property implies that even if the coalescence time of gene copies taken from different individuals is given by two distributions clearly differing in their means, the resulting combined distribution will not have two distinct peaks but will always have its highest value at the point farthest to the left. The second property implies that all estimates of coalescence times will have a very large standard error.

A slightly paradoxical effect of the results just described is that any measure of genetic diversity based on pair-wise comparisons of gene copies (such as the standard index of ‘heterozygozity’, *H*) will always be *higher* in populations dominated by asexuality than in otherwise similar sexual populations (given that the populations have reached their equilibria between mutation and drift). This follows from the fact that in asexual populations, individuals may carry gene copies that have not been given a chance to coalesce for a long time. As can be seen from Fig. 1, however, this effect becomes clearly noticable first when sexuality is very rare.

*n*-gene coalescence and diversity

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

Studying the coalescence behaviour of exactly *two* gene copies in the theoretical, disomic organism considered here involves a special situation, as these two copies can have coexisted in an asexual lineage for a long period of time. That this is a ‘special case’ from a mathematical point of view becomes clear when an analysis is made of the mean number of generations one needs to go back before *n* gene copies coalesce to a smaller number.

There is no simple formula for the *n*-coalescence under partial asexuality. Its basic properties, however, can be deduced by considering the two extreme cases, full sexuality and full asexuality.

In the first case, it is well known that the mean time to a coalescence event for a sample of *n* gene copies is 4*N*/*n*(*n –* 1) (see Gillespie, 1998, p. 41). In the second case, that of full asexuality, it can be shown (see appendix) that the corresponding value is 8*N*/*n*(*n –* 2) when *n* is an even number and 8*N*/(*n –* 1)(*n* + 1) when *n* is an uneven number. The mean time to an event of coalescence is thus about twice as great under conditions of asexuality than of sexuality. It should be noted, however, that the normal coalescence event under conditions of sexuality is one involving the transition from *n* to *n –* 1 independent gene copies; in the asexual case, the normal step is instead one from *n* copies to *n –* 2 copies. The mean time for coalescence from *n* copies down to two copies is thus similar in the two cases. It is only in the last step, from two copies to one, that the importance of sex becomes pronounced. If there is no sexuality in the population, this step can never even be made.

### Model 2. Genotypic variation under partial asexuality and drift

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

Attention will now be directed at the pattern of multilocus variability expected in populations of lesser size and more recent decent than those just considered. An index of genotypic identity, *GI*, will be used for describing the degree of variation in such populations. *GI*_{t} is the probability that two randomly sampled adult individuals from the population in generation *t* have the same genotype. *GI* can be interpreted as a measure of the degree of ‘clonality’ in the population. When *GI* is close to 1/*N*, each individual has its own unique multilocus genotype; this is the normal situation in outbreeding organisms without asexual reproduction. When *GI* is 1, all individuals in the population have the same genotype and belong to the same clone.

This index of genotypic identity, *GI*, is a version of the standard ‘identity by descent’ index for gene copies in populations (see e.g. Malécot, 1969) and can be studied by use of the same analytic tools. Most of the results given below are thus already known at least in principle (see e.g. Nei *et al*., 1975, regarding the relationship between population size and mutational divergence after population bottle-necks), but they are repeated here in an attempt to provide a coherent summary of the interplay between drift and sexuality/asexuality in limited populations. Particular attention is given to the conditions under which simplifying approximations apply; for a more detailed simulation of the dynamics of clonal diversity under different parameter assumptions, see Watkinson & Powell (1993).

The number of reproductive individuals in generation *t* is denoted as *N*_{t}; these individuals may contribute to the next generation sexually, asexually or both. The proportion of adults in generation *t* that are derived from sexual events is denoted as *σ*_{t} (unless otherwise specified, in which case no subscript appears, this parameter is treated as a constant). It is assumed that mutations can be ignored, which implies that asexually related individuals always share the same genotype. This assumption can easily be changed; it is included, however, because it simplifies the analysis and because only short time-scales will be considered. It will also be assumed that all sexual events lead to new, unique genotypes; this corresponds to what in a mutational context would be called an ‘infinite alleles’ assumption. As before, selection is ignored and all individuals are assumed to have identical reproductive systems.

A brief consideration will also be given to the genotypic relationship between two populations that share a common origin but which currently evolve independently. To this end, use will be made of another index, *BI*_{t}, defined as the probability that two individuals drawn at random in generation *t*, one from each population, will have the same genotype.

### The dynamics of genotypic identity

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

Letting *E*( ) denote the expected value of the variable in parenthesis, the relationship between the degrees of genotypic identity in two consecutive generations of a population can be written as

- (3)

Thus, the expected degree of clonal uniformity in a population depends on its value in the preceding generation (*GI*_{t-1}), the number of adult individuals from which the present population is derived (*N*_{t-1}), and – obviously enough – the level of sexuality in the organism (*σ*). In particular one can note that, irrespective of the degree of sexuality, a genotypically variable population remains variable if only a fair number of parents is used for deriving the next generation.

If between generations *T* + 1 and *T* + *t –* 1, the size of the population, *N*, remains the same, the expected value of the degree of genotypic identity in generation *T* + *t* can be written as

- (4)

where

- (5)

Thus, with time the degree of identity in the population will depend less and less on its exact value in the starting generation *T (*this follows from the factor with which *GI*_{T} is multiplied being less than unity). The rate at which this dependency decays is given by the largest of the two values 2*σ* and *N*^{−1}. This implies that after about 2 min [*N*, (2*σ*)^{−1}] generations of constant population size, the initial genotypic composition of the population can be ignored. Table 1 presents the expected genotypic identities in populations that have lived through a variable number of generations of constant size. The theoretically derived results concerning the times needed for populations to lose the ‘memories’ of their starting conditions are clearly illustrated.

Table 1. The expected genotypic identity, *E* ( *GI* ), calculated from expression (5), in a population consisting of *N* individuals for the last *t* generations. *GI*_{0} is the genotypic identity in the population at its start. A fraction σ of the individuals are sexually derived, the rest being asexually produced. *N* | *σ* | *t* | *E* ( *GI* ) |
---|

10 | 0.50 | 10 | 0.032 + 0.000 *GI*_{0} |

10 | 0.50 | 100 | 0.032 + 0.000 *GI*_{0} |

10 | 0.10 | 10 | 0.286 + 0.042 *GI*_{0} |

10 | 0.10 | 100 | 0.299 + 0.000 *GI*_{0} |

10 | 0.01 | 10 | 0.594 + 0.285 *GI*_{0} |

10 | 0.01 | 100 | 0.831 + 0.000 *GI*_{0} |

100 | 0.10 | 10 | 0.036 + 0.110 *GI*_{0} |

100 | 0.10 | 100 | 0.041 + 0.000 *GI*_{0} |

100 | 0.01 | 10 | 0.086 + 0.740 *GI*_{0} |

100 | 0.01 | 100 | 0.314 + 0.049 *GI*_{0} |

1000 | 0.10 | 10 | 0.004 + 0.120 *GI*_{0} |

1000 | 0.10 | 100 | 0.004 + 0.000 *GI*_{0} |

1000 | 0.01 | 100 | 0.041 + 0.121 *GI*_{0} |

1000 | 0.01 | 1000 | 0.047 + 0.000 *GI*_{0} |

If the population size remains the same for a long period of time, the expected degree of genotypic identity approaches an equilibrium value, *GI**, given by

- (6)

If *σ* is small and *N* is large, the following approximation holds

- (7)

This formula for the expected genotypic identity in a partially asexual population has been derived earlier, for example by Brookfield (1992) and Burt *et al*. (1996). Table 2 gives exact and approximate values for *GI**, from which can be seen that the approximation holds well already for *σ*≤0.10 and *N* ≥ 10.

Table 2. The expected equilibrium genotypic identity, *GI* * , in a population of fixed size, *N* . A fraction *σ* of the individuals are sexually derived, the rest being asexually produced. The exact value is calculated from expression (6), whereas the approximate value is obtained from expression (7). *N* | *σ* | *GI** |
---|

Exact | Approximate |
---|

10 | 0.50 | 0.032 | 0.091 |

10 | 0.20 | 0.151 | 0.059 |

10 | 0.10 | 0.299 | 0.333 |

10 | 0.05 | 0.481 | 0.500 |

10 | 0.01 | 0.831 | 0.833 |

100 | 0.20 | 0.017 | 0.024 |

100 | 0.10 | 0.041 | 0.048 |

100 | 0.05 | 0.085 | 0.091 |

100 | 0.01 | 0.330 | 0.333 |

100 | 0.005 | 0.498 | 0.500 |

100 | 0.001 | 0.833 | 0.833 |

1000 | 0.10 | 0.004 | 0.005 |

1000 | 0.01 | 0.047 | 0.048 |

1000 | 0.001 | 0.333 | 0.333 |

### Density dependent growth and establishment

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

Of the many different aspects of Model 2 that can be elaborated in search of better fits to actual situations, only one will be considered here: the effect of population *density* on the demography and breeding system of the population. Many populations start from a small number of founders. They then develop in size – sometimes slowly, sometimes explosively – during which interindividual competition becomes more and more important. Finally, an equilibrium population size is reached that is maintained until the population – slowly or abruptly – becomes extinct.

To illustrate the effect of the rate of population growth immediately after the population is being founded, the degree of genotypic identity in populations of differing starting conditions and population parameters was investigated numerically by computer simulations. In all cases, the population development was assumed to follow a logistic growth curve. Thus, in expression (4) above one lets

- (9)

where *r* and *K* denote the unlimited growth rate and the carrying capacity of the population, respectively.

In numerical iterations this recursion was followed deterministically over time with rounded values of *N*_{t} being used for the calculation of consecutive values of *E*(*GI*_{t}). Two such evolutionary runs, based on expressions (3) and (9), are shown in Fig. 3. It can be seen there that in a population started from a small set of founders, slow population growth leads initially to an increase in the degree of clonality, but that this effect gradually disappears as the population develops. In an organism that has rapid growth, this bottleneck effect is less important.

When the degree of competition between the individuals in the population changes, it is likely that the rate of establishment of sexually derived propagules is also affected. Sexually derived offspring are often smaller than asexually derived ones and in dense stands they find fewer opportunities to become established. This effect can be modelled by letting the proportion of sexually derived adults in a population of *N*_{t} individuals be

- (10)

where *σ* is the rate of establishment of sexual individuals when no competition is present and *β* is a measure of how sensitive the establishment of sexual offspring is to population density. In this model of *σ*_{t}, the role of sexuality in the population decreases towards zero as the population size approaches its equilibrium value.

An illustrative run is shown in Fig. 4. The simple additional factors considered – a logistically growing population and a density-dependent rate of establishment of sexually derived offspring – cause the population to display widely different levels of genotypic identity during its investigated history. Although many different numerical examples of this variability could be provided, they do not contribute anything further to the general point, that it is very difficult to judge the ‘effective level of sexuality/asexuality’ in a population from data derived from a single generation. Such a procedure will normally only work if the interacting factors of population dynamics, drift and sexuality from other evidence are known to be in equilibrium.

### Discussion

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

In considering the variability in a partially asexual organism, one needs to take account of the relevant time scale.

If the organism has had its present breeding system for a long period of time, the results of Model 1 are of relevance. Organisms with a long evolutionary history of asexual reproduction should contain highly divergent genetic material (Birky, 1996; Normark, 1999), a suggestion that recently has been empirically exemplified (Welch & Meselson, 2000). The primary aim of the present study was to quantify the *amount* of asexuality needed for an organism to turn its allelic variability into this asexual pattern. As seen in Fig. 1, there needs to be less than one sexual event about every second or third generation for the characteristic pattern of highly divergent gene copies within individuals to evolve.

This result has a number of implications. The first is that asexuality does not decrease genetic variation in the sense of divergence between gene copies, its effect if anything being the opposite of this. The mean time to coalescence for two randomly sampled gene copies is longer on average in partially asexual populations than in sexual populations that otherwise are similar. This effect is only of measurable size, however, in highly asexual populations.

Secondly, one may well find organisms that from a phenotypic point of view appear completely asexual but which nevertheless fail to show the characteristic pattern of asexual allelic divergence. As was seen above, only a few sexual events per generation is sufficient to prevent this genomic pattern from developing, and such rare events may easily go unnoticed.

The third implication is that when rare sexual events become as important as they are in the present context, even ‘unconventional’ recombinational events can play an important role. Such a possibility has been proposed, for example by Normark (1999), who suggested that the absence of extensive allelic differences in a supposedly asexual species of aphids could be due to rare cases of mitotic recombination.

It can also be noted, that one of the key assumption in Model 2 – that all sexual events lead to new multilocus genotypes – becomes reasonable when considered in the light of the results obtained with Model 1. As the expected degree of variation between gametes from predominantly asexual organisms is high, every sexual event that occurs produces a new, unique multilocus genotype. The only situation in which this argument fails to hold is that of recent asexual lineages being derived from strict selfers; cases of this sort, however, being rare in nature. Thus, if a good system for the screening of genetic markers is employed, it is highly unlikely that a new clone derived from a sexual event will fail to be recognized, irrespective of the exact origin and breeding system of the population.

When attention is shifted from the level of allelic variation in an organism to its genotypic variation, it is natural to shift the time scale too. Now the time horizon is not given by the breeding system of the organism during its last thousand or million generations, but by the reproductive processes in the present population that may have existed for only few years. Also for this short time scale, it turns out to be the *number* of sexually derived individuals per generation –*σN* or *s*– that is decisive for the expected pattern of variation. If some sexual individuals are formed in each generation, a fair amount of clonal variation is expected to be found in the population, whereas if there is less than one sexual event per generation, monomorphism may well occur [see expressions (7) and (8) and Table 3].

For simplicity, the role of mutations has been ignored in the formulation of Model 2. However, if many genetic markers are studied and each has a reasonably high mutation rate, it may well be that this assumption becomes too restrictive (which occurs when the rate of multilocus mutation becomes greater than the rate of sexuality). Equations (3)–(12) above are still relevant nevertheless. The only change in them needed in order to take mutations into account is to replace *σ* throughout by *σ* + *μ*, where *μ* is the multilocus mutation rate.

It is of interest to study Models 1 and 2 jointly under the reasonable assumption that the organism has a low but not negligible degree of sexuality and that its population structure is given by a large metapopulation in which small, isolated and relatively short-lived subpopulations are constantly being formed and lost. Under such conditions one would expect to find almost no genotypic variation within subpopulations, at the same time as the pattern of allelic divergence would not show the typical signs of long-term, strict asexuality. In addition, expression (12) indicates that even if on average the subpopulations will show little or no within-subpopulation variation, their relative between-population variation will increase constantly during the period of their existence. The resulting picture will, thus, appear confusing: there being many divergent subpopulations which show little internal clonal diversity, at the same time as there are no signs of long-term asexuality in the divergence between gene copies within individuals.

In general, the results of Model 2, as illustrated for example in Fig. 3 and 4, indicate that predominantly asexual populations can display almost any pattern of genotypic variation conceivable. From the very fact of asexuality occurring in a population (together with the absence of selection), it follows that the population possesses an effective ‘memory’ of its earlier genetic history. For this reason, it will take many generations before a predominantly asexual form derived from a single, unique event will show any clear signs of clonal divergence because of its sexuality, particularly if its population size is restricted (*Rubus scheutzii*, as described by Kraft & Nybom, 1995; can serve to illustrate this). Such a ‘memory-effect’ can explain more than simply the absense of variation. A population which was started by a number of sexually derived propagules may thus retain its initial genotypic variation for a very long period of time, even if it later reproduces almost exclusively asexually. A reasonably large population size is sufficient to ensure this effect. A large population size also in itself fosters genotypic variation, provided the time available is sufficient, even when sexuality is rare. This follows, for example, from expression (7). There is thus rarely any reason to invoke selection, somatic mutations, or any other specific mechanism to explain the presence of genotypic variations in organisms that are phenotypically highly asexual but retain the capacity to produce sexual offspring (an example of this may be *Carex bigelowii*, no seed-derived plantlets of which have been observed in natural stands, at the same time as it is highly variable genotypically; see Jonsson *et al*., 1996).

The reason for there being a balance between asexual and sexual reproduction in an organism is often difficult to understand (see, e.g. Bengtsson & Ceplitis, 2000). As shown here, it is often also difficult on the basis of population genetics data to determine whether any such balance *exists*, as – depending on when and how population information is obtained – an organism may appear almost fully sexual or strictly asexual. The best approach to estimating past and current breeding systems is probably to use a combination of different kinds of data, relating for example allelic variability to genotypic variability, or within-population divergence to between-population divergence. Methods of this sort, however, require more specific modelling than was performed here, both with respect to the kind of data employed (information about more than one gene copy per individual is often available, for example) and the assumptions that can be made about the organism's population dynamics. The way in which direct and indirect selective processes influence the pattern of population variability must also be taken into account. In the present study, the attempt was made to describe the interactions between the most important processes that affect neutral genetic variation in partially asexual organisms, and to do so in as general a setting as possible. The framework thus produced should be able to serve as a kind of null-model for more detailed future studies.

where **G** is the 2 × 2 matrix having the *g*-values as elements and **c** is the column vector having the *c*-values as elements. If the probability of two gene copies currently in the same individual having coalesced *t* generations back is denoted by *P*_{2}(*t*), then

- ((A2))

The present analysis is limited to the situation in which the number of sexual events per generation, *s*, is fixed, i.e. when the rate of sexuality *σ* = *s*/*N* is of the order of *N*^{−1}. The results for larger values of *σ* can be obtained by letting *s* grow towards *N* in the equations below.

The matrix **G** has two eigenvalues which with relevant approximations can be written as

- ((A3))

where *α*_{1} and *α*_{2} are as given in the text (expressions 2 and 1, respectively). Using spectral theory and some straightforward calculations, one obtains

- ((A4))

where

- ((A5))

One can see here how both coalescence probabilities can be written as linear combinations of two processes which, for large values on *N* and *t*, approximate geometric distributions having means of *N*/*α*_{1} and *N*/*α*_{2}. The relative importance of the two processes for extreme values of *s* is shown by the following relationships:

A verbal summary of these results is given in the text.

It is straightforward to extend these results to take inbreeding in the form of selfing into account. Using the notation introduced in the text, the elements of **G** become: *g*_{11} = *σo*(1 − 1/*N*), *g*_{12} = 1 − *σ* + *σ*/2*N* + *σ*(1 − *o*)/2, *g*_{21} = *σo*(1 − 1/*N*) and *g*_{22} = 1 − *σ*[1 − *o*/2*N –* (1 − *o*)/2]. The eigenvalues of **G** and the *P*-values are as in (A3–A5 above) except that now

and

- ((A6))

*n*-gene coalescence under no or complete asexuality

- Top of page
- Abstract
- Introduction
- Model 1. Allelic variation under mutation, drift and partial asexuality
- 2-gene coalescence and diversity
*n*-gene coalescence and diversity- The effect of inbreeding
- Model 2. Genotypic variation under partial asexuality and drift
- The dynamics of genotypic identity
- Expected number of clones in a sample
- Density dependent growth and establishment
- Genotypic variability between populations
- Discussion
- Acknowledgments
- References
- Appendix: coalescence calculations under partial asexuality
*n*-gene coalescence under no or complete asexuality

Without asexuality, i.e. when *σ* = 1, a standard result is that the *n*-gene coalescence process has a mean of 4*N*/*n*(*n –* 1) (see Gillespie, 1998, p. 41).

When asexuality is involved, one needs to take account of the fact that a sample of *n* gene copies can be differentially distributed between the individuals in a population. The copies can be found in *n* different individuals (state 1); alternatively, *n –* 2 copies can be found in that same number of individuals, at the same time as two distinct copies can be found in the same individual (state 2); and so on down to the state in which each of *n*/2 individuals contains two of the original copies (for the case of *n* being even), or each of (*n –* 1)/2 individuals contains two copies and one individual contains a single copy. Denote the total number of such states as *k*. Each state *i* is associated with a probability *c*_{i} that in the preceding generation the *n* gene copies were fewer due to coalescence. Each state *i* is also associated with a probability *g*_{ij} that the population in the preceding generation was in state *j*. Using this notation, the probability of a coalescence event having occurred exactly *t* generations back can be written in the same form as (A2).

When *N* is large, the matrix **G** given by the *g*_{ij} values has strictly positive elements located on, just above, and just below the main diagonal, no account needing to be taken of the other elements. Instead of specifying the retained elements and trying to find an analytical form for the largest eigenvalue of **G**, it is for our purpose sufficient to consider what happens when asexuality increases towards 1 (σ 0). There will then be no significant elements below the diagonal, all of them being either on or above it. The eigenvalues will be identical to the elements located on the diagonal, and it is easily shown that these are

- ((A7))

Since the eigenvalues increase in size with *i*, the largest eigenvalue is 1 − *n*(*n –* 2)/8*N* for an even value of *n*, and 1 − (*n –* 1)(*n* + 1)/8*N* for an uneven value of *n*. As *n* increases, both values come to approximate 1 − *n*^{2}/8*N*. Note that this largest eigenvalue is associated with the state in which the *n* gene copies occur in as few individuals as possible, a state which implies that when a coalescence event occurs not one but two gene copies will coalesce at the same time. When *n* equals 2, the largest eigenvalue equals 1, which implies that no coalescence can occur. This result reflects the fact that in the absence of sexuality two separate gene copies in a single individual cannot coalesce.