Abstract
Using the transition matrix of inbreeding and coancestry coefficients, the inbreeding (N_{eI}), variance (N_{eV}), and asymptotic (N_{eλ}) effective sizes of mixed sexual and asexual populations are formulated in terms of asexuality rate (δ), variance of asexual (C) and sexual (K) reproductive contributions of individuals, correlation between asexual and sexual contributions (ρ_{ck}), selfing rate (β), and census population size (N). The trajectory of N_{eI} toward N_{eλ} changes crucially depending on δ, N, and β, whereas that of N_{eV} is rather consistent. With increasing asexuality, N_{eλ} either increases or decreases depending on C, K, and ρ_{ck}. The parameter space in which a partially asexual population has a larger N_{eλ} than a fully sexual population is delineated. This structure is destroyed when N(1 –δ) < 1 or δ> 1 –1/N. With such a high asexuality, tremendously many generations are required for the asymptotic size N_{eλ} to be established, and N_{eλ} is extremely large with any value of C, K, and ρ_{ck} because the population is dominated eventually by individuals of the same genotype and the allelic diversity within the individuals decays quite slowly. In reality, the asymptotic state would occur only occasionally, and instantaneous rather than asymptotic effective sizes should be practical when predicting evolutionary dynamics of highly asexual populations.
UNIPARENTAL reproduction by either asexual reproduction or selffertilization would be vitally important for sedentary organisms to survive at a low population density that may occur when the populations are exploring a new habitat or recovering from destructive disturbance. Of these two systems of uniparental reproduction, asexual reproduction has the advantage of saving resources for reproduction and retaining the heterozygosity of individuals. In fact, many plants and lower animals reproduce fully or partially asexually (Harper 1977; Kawano 1984; Richard 1986; Smith and Szathmary 1999). In spite of this common occurrence of asexuality, population genetic structure of asexually reproducing species has not been well studied. A group of individuals with any rate of asexuality (excluding perfect asexuality) constitutes a reproductive community, i.e., Mendelian population, and should have a particular genetic or evolutional structure of its own. We need to elucidate this structure to extend our knowledge on the dynamics of biological evolution.
Both stochastic and deterministic forces determine the evolutional trajectory of a population. The stochastic aspect of evolution depends on the wellknown parameter, effective population size N_{e}, a concept that was coined by Wright (1931, 1969) to standardize the actual population to an ideal one and has been widely used as a key parameter in predicting the evolutional dynamics of finite populations (Crow and Kimura 1970; Hartl and Clark 1989; Caballero 1994; Wang and Caballero 1999) as well as in solving various optimization issues in population management (Franklin 1980; Soulé 1980; Lande 1995; Santiago and Caballero 1995). Various kinds of N_{e} have been defined, such as for inbreeding, variance, and eigenvalue, denoted here N_{eI}, N_{eV}, and N_{eλ}, respectively (Ewens 1982; Crow and Denniston 1988). These sizes determine the progress of inbreeding (identity by descent of two allelic genes within individuals), the magnitude of random drift in gene frequency, and the asymptotic rate of decay of segregating loci, respectively. N_{eI} and N_{eV} are not identical in general, but asymptote to a common N_{eλ} with increasing generations in sexually (but not fully selfing) reproducing populations (Ewens 1979; Pollak 1987; Chesseret al. 1993). Therefore, these three kinds of N_{e} are the same in populations that have persisted for sufficiently many generations.
Asexual reproduction prevents the advance of inbreeding and may modify the reproductive pattern of individuals and consequently will change the dynamics as well as the eventual, asymptotic value of N_{eI} and N_{eV}. To discuss this issue, each effective size must be formulated in terms of the rate of asexuality as well as some other parameters specifying the reproductive pattern of individuals. Orive (1993) and Balloux et al. (2003) addressed this issue, using the coalescence time theory. However, they did not consider some important reproductive parameters and discussed only asymptotic effective sizes. These parameters were incorporated in our N_{e} (Yonezawa 1997; Yonezawaet al. 2000), which, however, was also asymptotic. In this article, we formulate both instantaneous and asymptotic effective sizes, using the transition matrix of the inbreeding and coancestry coefficients, and discuss how these N_{e}'s are modified by asexuality as well as by other reproductive parameters.
FORMULATIONS OF THE EFFECTIVE POPULATION SIZE
A population of a diploid monoecious (or hermaphroditic) species with discrete generations and constant census size N is discussed. Individuals in the population are assumed to have genetically the same capacity of reproduction (progeny size) and rate of asexuality (δ), although phenotypically these parameters are subject to chance fluctuation due to some nongenetic factors such as environmental heterogeneity within population and sampling accidents that occur when sexual and asexual propagules are chosen for the next generation. The asexuality may be of any kind including apomixis (reproduction by unreduced egg cells). Each individual does selfing with a rate β. Individuals produced asexually (referred to as asexual individuals in what follows) are assumed to have the same reproductive patterns as those produced sexually (sexual individuals).
Assuming neutrality of genes and absence of mutations, the coefficient of inbreeding in generation t, denoted f_{t}, can be described in terms of the coefficients of inbreeding and coancestry of the preceding generation as
The coancestry θ_{t} is obtained as a sum of three components θ_{1}_{t}, θ_{2}_{t}, and θ_{3}_{t}, of different causations, i.e., between two asexual individuals, between one asexual and one sexual individual, and between two sexual individuals, respectively. The first component θ_{1}_{t} is given by
Symbol ρ_{ck} comprising X_{1} stands for the correlation coefficient between the asexual (c_{i}) and sexual (k_{i}) reproductive contributions. The reproductive correlation has been commonly observed in plant species (Harper 1977; Kawano 1984; Waller 1988) and was newly incorporated in this article.
The third component is presented as
From Equations 1, 3, 5, and 7, the transition matrix of inbreeding is obtained as
INFLUENCE OF ASEXUAL REPRODUCTION ON THE EFFECTIVE POPULATION SIZE
Instantaneous effective size: N_{eI,}_{t} and N_{eV,}_{t}, under a Poissondistributed asexual and sexual reproductive contribution (C = 1 and K = 1 +β) with the initial condition f_{0} =θ_{0} = 0, were calculated for some typical values of the parameters involved. As illustrated in Figure 1, the trajectory of N_{eI,}_{t} differs remarkably with different rates of asexuality as well as selfing. It also depends on the census population size N as shown later. Such a large change in the trajectory of N_{eI,}_{t} occurs because the initial size, N_{eI,1}, which is obtained as 1/{(1 –δ)β} from the transition matrix (8), takes quite different values depending on the values of δ and β.
As generations proceed, N_{eI,}_{t} asymptotes either downward or upward to the eigenvalue effective size N_{eλ}, depending on whether N_{eI,1} is larger or smaller than N_{eλ}. N_{eI,}_{t} is constant over generations when N_{eI,1}equals N_{eλ}. A critical value of δ, denoted
Asymptotic effective size: It is seen from Equation 10 that the asymptotic effective size N_{eλ} is the same at any rate of asexuality when C = (K + 1 –β)/2 and ρ_{ck} = 0. The condition C = (K + 1 –β)/2 holds under Poissondistributed reproductive contribution (C = 1 and K = 1 +β). Otherwise N_{eλ} changes with increasing asexuality rates in different patterns depending on the values of C, K, and ρ_{ck} (Figure 2). When ρ_{ck} = 0, the relative magnitude of C to (K + 1 –β)/2 alone counts; N_{eλ} increases with increasing δ when C < (K + 1 – β)/2 (case 2 in Figure 2), whereas it decreases when C > (K + 1 –β)/2 (case 3). This trend is considerably modified in the presence of a reproductive correlation; ρ_{ck} acts to enlarge N_{eλ} when negative and to decrease it when positive.
The effect and interrelationship of the various reproductive parameters can be more comprehensively grasped on the basis of the term
Highly asexual and small populations: When the terms of the second as well as the first order of 1/N are considered, N_{eλ} is formulated as
This effect of high asexuality can be grasped in a different way. Without asexuality, the asymptotic effective size is mostly proportional to the census population size N (Caballero 1994; Wang and Caballero 1999), and then the effective to census size ratio N_{eλ}/N is constant over changing N. It is known by Equation 13 that the constancy of N_{eλ}/N is destroyed when δ> 1 – 1/N. The calculations of Figure 5 show that with an asexuality rate as high as 0.99, N_{eλ}/N increases markedly with decreasing N. It may be said that asexuality has a buffering effect of stabilizing N_{eλ} against decreasing N.
The mechanism of the abovementioned effect of asexuality can be explicitly interpreted when considering the dynamics of N_{e} of a fully asexual population (δ= 1). Inbreeding does not progress under complete asexuality, and therefore N_{eI,}_{t} is always infinite. With f_{0} = θ_{0} = 0, N_{eV,}_{t} is derived from Equation 8 as
With an extremely high if not perfect asexuality, the population eventually should have a similar genetic structure as mentioned above, being overwhelmed by copies of an ancestral genotype. The initial genetic diversity should decay more rapidly and thoroughly in smaller populations, explaining why the sudden increase in N_{eλ} (Figure 4) occurs more markedly with smaller populations. However, in contrast to the case of perfect asexuality, the allelic diversity within individuals continues to decay, although quite slowly. Then, the asymptotic effective size should be extremely large. This large N_{eλ} indicates the persistence of rather than the richness of alleles.
In highly asexual populations, the concept of asymptotic size would not be practical, although important theoretically. With a high asexuality, the asymptotic state is established very late; hundreds of generations are required for the asymptotic state to be established after foundation of a population or for it to be recovered once disturbed (Figure 6). In reality, therefore, the asymptotic state would occur only occasionally. In this situation, instantaneous rather than asymptotic effective sizes should be practical. When discussing the evolutionary dynamics over t generations, the harmonic mean of instantaneous effective sizes of these generations should be used. By the relation
The variance harmonic mean N_{eVh} is stable over many generations; e.g., in Figure 7, N_{eVh} is practically constant over hundreds of years (although it increases gradually when N < ∼200). Therefore, an instantaneous variance effective size that was estimated initially, i.e., N_{eV,1}, can be used over sufficiently many generations that follow. From Equation 8, N_{eV,1} is formulated as
In contrast to variance effective size, inbreeding effective size decreases toward N_{eλ} as generations proceed (Figure 6). Reflecting this (note that the value of harmonic mean depends strongly on the smallest component), N_{eIh} decreases notably with increasing generations, although it is almost constant under a high selfing rate (Figure 7). Therefore, an instantaneous inbreeding effective size estimated in the first generation, N_{eI,1}, may not be used for multiplegeneration prediction.
To summarize, against our intuitive thought, asexuality is not always a negative factor that decreases the genetic diversity of populations. It acts either negatively or positively depending on the various reproductive parameters, especially the magnitude of the asexual reproductive variance (C) relative to the sexual one (K). With a sufficiently uniform asexual contribution of individuals (small C) and negative reproductive correlation (ρ_{ck} < 0), asexuality could be highly advantageous to retain genetic diversity (Figure 2). In plant species, highly asexual populations very often show poorer genetic diversities than sexual populations (Pleasants and Wendel 1989; Aspinwall and Christian 1992; Pellegrin and Hauber 1999). There are many possible scenarios of this phenomenon; the original genetic diversity may have been lost due to a large asexual reproductive variance (large C and small N_{e}), or the asexual populations may have been founded by only one or a few genotypes (a founder effect or extinctionrecolonization dynamics), or data of the asexual populations may have been collected from only a few among many asexual lineages (biased sampling). Moreover, in highly asexual populations, the original as well as mutational genetic diversities may be rapidly lost under pressure of natural selection (Ballouxet al. 2003). To identify the true scenario, not only the allelic and genotypic diversities of the populations but also various parameters concerning reproductive pattern and selection as well as mutations must be known.
DISCUSSION
Extension to generationoverlapping populations: Many of the partially or fully asexually reproducing
plant species are perennial (Harper 1977; Kawano 1984; Waller 1988). Populations in these species are composed of individuals of different ages and, in some species, of different demographic stages. N_{eI} and N_{eV} for these populations can be derived straightforwardly if the yearly (or seasonal) change in f_{t} and θ_{t} is formulated in a form like Equation 8. Then, similar mathematical procedures as used above give the annual effective sizes (Hill 1972) for f_{t} and θ_{t}, denoted N_{yI,}_{t} and N_{yV,}_{t}, respectively, both of which asymptote to a common asymptotic annual effective size N_{yλ}. The lifetime inbreeding effective size, denoted N_{eI,}_{T}, can be defined as (1 – f_{T}_{–1})/{2(f_{T} – f_{T}_{–1})}, where T is counted in units of the demographic generation length L, i.e., the mean age at which new individuals are produced either sexually or asexually (Orive 1993). Because a relation
Conservation genetic interpretation: The reproductive pattern (C, K, and ρ_{ck}) can be controlled artificially to enlarge N_{e} with the census size N unchanged. A population that reproduces mixed sexually and asexually in nature may be conserved fully asexually or sexually with the same number of progeny being propagated from each individual sampled. Three of the most practicable propagation schemes and N_{e}'s under these schemes are formulated in Table 1; in scheme I, the population is propagated fully asexually, while in schemes II and III it is propagated fully sexually without or with control of outcrossing, respectively. In formulating the N_{e}'s, the initial population was assumed to have an asymptotic genetic structure, having a deviation of β/(2 –β) from the HardyWeinberg proportions. In this case, N_{e} in each scheme can be derived from Equation 10 with the reproductive parameters C, K, and ρ_{ck} being defined as presented in Table 1.
The calculations illustrated in Figure 8 show that the relative effectiveness of these schemes depends on the sampling fraction u. When more than half of the individuals are sampled to produce progeny (u > 0.5), scheme I is superior to II and III, whereas II is the best when less than half of the individuals (u < 0.5) are sampled in mainly outcrossing populations. Scheme III is superior to II only with sampling fractions as high as or higher than ∼0.8. The difference of the three schemes diminishes as β gets close to unity; N_{e} of all three equals uN/(1 – u) when β= 1.
Coalescence effective size: Orive (1993) and Balloux et al. (2003) formulated N_{e} of both sexual and asexual populations, using the coalescence time theory. This approach has some advantages, such as providing exact analytical expressions and defining both allelic and genotypic effective size (L. Lehmann and F. Balloux, unpublished results). However, the reproductive parameters C, K, and ρ_{ck} are difficult to incorporate explicitly and instantaneous N_{e} cannot be known by the coalescence time theory.
Balloux et al. (2003) showed that asexuality causes no noticeable change in N_{e} unless occurring at an extremely high rate, and N_{e} suddenly increases toward infinity when the asexuality rate tends toward unity. By our equations, N_{eλ} either increases or decreases with increasing asexuality, depending on the reproductive parameters C, K, and ρ_{ck} (Figures 2 and 4). The trend pointed out by Balloux et al. (2003) occurs when C = 1, K = 1 +β, and ρ_{ck} = 0.
The calculations of coalescence time of Bengtsson (2003) showed that partially asexual populations have the same pattern of allelic and genotypic variation as fully sexual populations as far as a few or more sexual individuals (cf. his Figure 1) are produced per generation. In the context of our approach, his finding can be expressed as asexuality (δ) causes no significant change in N_{eλ} when the number of sexual individuals N(1 –δ) is larger than a few. The term of the second order of 1/N in our Equation 13 can be ignored when N(1 –δ) is larger than a few, and then N_{eλ} can be approximated by Equation 10. By the calculations of this equation (Figure 2), the constancy of N_{eλ} over changing δ and therefore the finding of Bengtsson (2003) hold when C = 1, K = 1 +β [more generally, C = (K + 1 –β)/2], and ρ_{ck} = 0. It follows that both of the trends pointed out by Balloux et al. (2003) and Bengtsson (2003) are true under Poissondistributed (C = 1 and K = 1 +β) and independent (ρ_{ck} = 0) reproductive contribution of individuals.
This ideal pattern of reproductive contribution may hold when the reproductive variances C and K occur by only sampling accidents, but not when environmental heterogeneity exists within the population. In sedentary organisms, the fecundity as well as asexuality of individuals may vary, depending on the environmental conditions. In plants in particular, heterogeneity in some microenvironmental factors such as soil fertility, temperature, humidity, and solar radiation would cause a large change in the reproductive parameters of individuals, and then C and K could be much larger than one. This influence of environmental heterogeneity should be prominent especially in perennials. In populations of perennials, an individual that occupies a more fertile position produces more progeny each year and persists longer, leading to a large variation in the lifetime reproductive contribution of individuals (Yonezawa 1997). Meanwhile, the variances C and K can be minimized by an appropriate artificial management (cf. Table 1), thereby enhancing the effective population size.
Acknowledgments
The authors are greatly indebted to Dr. F. Balloux, who kindly sent us his manuscripts of related topics and gave valuable comments and suggestions.
Footnotes

Communicating editor: T. H. D. Brown
 Received June 16, 2003.
 Accepted December 10, 2003.
 Copyright © 2004 by the Genetics Society of America