Abstract
Using the transition matrix of inbreeding and coancestry coefficients, the inbreeding (NeI), variance (NeV), and asymptotic (Neλ) 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 (ρck), selfing rate (β), and census population size (N). The trajectory of NeI toward Neλ changes crucially depending on δ, N, and β, whereas that of NeV is rather consistent. With increasing asexuality, Neλ either increases or decreases depending on C, K, and ρck. The parameter space in which a partially asexual population has a larger Neλ 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 Neλ to be established, and Neλ is extremely large with any value of C, K, and ρck 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 self-fertilization 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 well-known parameter, effective population size Ne, 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 Ne have been defined, such as for inbreeding, variance, and eigenvalue, denoted here NeI, NeV, and Neλ, 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. NeI and NeV are not identical in general, but asymptote to a common Neλ with increasing generations in sexually (but not fully selfing) reproducing populations (Ewens 1979; Pollak 1987; Chesseret al. 1993). Therefore, these three kinds of Ne 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 NeI and NeV. 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 Ne (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 Ne'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 ft, can be described in terms of the coefficients of inbreeding and coancestry of the preceding generation as
The coancestry θt is obtained as a sum of three components θ1t, θ2t, and θ3t, 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 θ1t is given by
Symbol ρck comprising X1 stands for the correlation coefficient between the asexual (ci) and sexual (ki) 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: NeI,t and NeV,t, under a Poisson-distributed asexual and sexual reproductive contribution (C = 1 and K = 1 +β) with the initial condition f0 =θ0 = 0, were calculated for some typical values of the parameters involved. As illustrated in Figure 1, the trajectory of NeI,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 NeI,t occurs because the initial size, NeI,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, NeI,t asymptotes either downward or upward to the eigenvalue effective size Neλ, depending on whether NeI,1 is larger or smaller than Neλ. NeI,t is constant over generations when NeI,1equals Neλ. A critical value of δ, denoted
Asymptotic effective size: It is seen from Equation 10 that the asymptotic effective size Neλ is the same at any rate of asexuality when C = (K + 1 –β)/2 and ρck = 0. The condition C = (K + 1 –β)/2 holds under Poisson-distributed reproductive contribution (C = 1 and K = 1 +β). Otherwise Neλ changes with increasing asexuality rates in different patterns depending on the values of C, K, and ρck (Figure 2). When ρck = 0, the relative magnitude of C to (K + 1 –β)/2 alone counts; Neλ 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; ρck acts to enlarge Neλ when negative and to decrease it when positive.
—Asymptotic patterns of inbreeding (NeI,t) and variance (NeV,t) effective population sizes (in ratio to census size N) under some typical rates of asexual reproduction (δ) and selfing (β), calculated with specifications N = 100, f0 =θ0 = 0, ρck = 0, C = 1, and K = 1 +β. The half-life generations tI,0.5 and tV,0.5 were calculated by tI,0.5 = 1 + log Q/log(λ2/λ1), where Q = {2(1 –λ2) – (1 –δ)β}{1 –λ1 – 1/(2N̄eI)}/[{2(1 –λ1) – (1 –δ)β}{1 –λ2 – 1/(2N̄eI)}], N̄eI = (Neλ + NeI,0)/2, and λ2 equals the second eigenvalue of the matrix [Tij], and tV,0.5 = 1 + log Q′/log(λ2/λ1), where Q′= (T21 + T22 –λ2){1 –λ1 – 1/(2N̄eV)}/ [(T21 + T22 –λ1){1 –λ2 – 1/(2N̄eV)}], and N̄eV = (Neλ + NeV,0)/2.
The effect and interrelationship of the various reproductive parameters can be more comprehensively grasped on the basis of the term
—Effect of asexuality rate on Neλ under different patterns of reproductive contribution of individuals, calculated with β= 0.
Highly asexual and small populations: When the terms of the second as well as the first order of 1/N are considered, Neλ 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 Neλ/N is constant over changing N. It is known by Equation 13 that the constancy of Neλ/N is destroyed when δ> 1 – 1/N. The calculations of Figure 5 show that with an asexuality rate as high as 0.99, Neλ/N increases markedly with decreasing N. It may be said that asexuality has a buffering effect of stabilizing Neλ against decreasing N.
The mechanism of the above-mentioned effect of asexuality can be explicitly interpreted when considering the dynamics of Ne of a fully asexual population (δ= 1). Inbreeding does not progress under complete asexuality, and therefore NeI,t is always infinite. With f0 = θ0 = 0, NeV,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 Neλ (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 Neλ indicates the persistence of rather than the richness of alleles.
—Delineation of the condition for asexuality causing a positive or negative effect on the effective size (Neλ). The intercept of the line is (1 –β)/2. See text for the interpretation of this figure.
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
—Effect of asexuality rate on Neλ when calculated by the exact eigenvalue effective size. Calculated with β= 0 and ρck = 0.
—Ratio of the effective to census population size (Neλ/N) under varying census population sizes (N). Neλ was calculated using 1/{2(1 –λ1)} with specifications β= 0, C = K = 1, and ρck = 0.
—Asymptotic patterns of inbreeding and variance effective sizes under extremely high rates of asexuality, calculated with N = 100, f0 =θ0 = 0, β= 0, C = K = 1, and ρck = 0.
The variance harmonic mean NeVh is stable over many generations; e.g., in Figure 7, NeVh 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., NeV,1, can be used over sufficiently many generations that follow. From Equation 8, NeV,1 is formulated as
—Harmonic mean over t generations of the instantaneous inbreeding and variance effective sizes under extremely high rates of asexuality, calculated with N = 500, f0 =θ0 = 0, C = 1, K = 1 +β, and ρck = 0.
In contrast to variance effective size, inbreeding effective size decreases toward Neλ as generations proceed (Figure 6). Reflecting this (note that the value of harmonic mean depends strongly on the smallest component), NeIh 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, NeI,1, may not be used for multiple-generation 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 (ρck < 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 Ne), or the asexual populations may have been founded by only one or a few genotypes (a founder effect or extinction-recolonization 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 generation-overlapping 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. NeI and NeV for these populations can be derived straightforwardly if the yearly (or seasonal) change in ft and θt is formulated in a form like Equation 8. Then, similar mathematical procedures as used above give the annual effective sizes (Hill 1972) for ft and θt, denoted NyI,t and NyV,t, respectively, both of which asymptote to a common asymptotic annual effective size Nyλ. The life-time inbreeding effective size, denoted NeI,T, can be defined as (1 – fT–1)/{2(fT – fT–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
Effective population sizes under three typical schemes of controlled propagation
Conservation genetic interpretation: The reproductive pattern (C, K, and ρck) can be controlled artificially to enlarge Ne 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 Ne'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 Ne's, the initial population was assumed to have an asymptotic genetic structure, having a deviation of β/(2 –β) from the Hardy-Weinberg proportions. In this case, Ne in each scheme can be derived from Equation 10 with the reproductive parameters C, K, and ρck 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; Ne of all three equals uN/(1 – u) when β= 1.
Coalescence effective size: Orive (1993) and Balloux et al. (2003) formulated Ne 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 ρck are difficult to incorporate explicitly and instantaneous Ne cannot be known by the coalescence time theory.
Balloux et al. (2003) showed that asexuality causes no noticeable change in Ne unless occurring at an extremely high rate, and Ne suddenly increases toward infinity when the asexuality rate tends toward unity. By our equations, Neλ either increases or decreases with increasing asexuality, depending on the reproductive parameters C, K, and ρck (Figures 2 and 4). The trend pointed out by Balloux et al. (2003) occurs when C = 1, K = 1 +β, and ρck = 0.
—Ne under the three propagation schemes I, II, and III in response to varying sampling fractions (u).
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 Neλ 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 Neλ can be approximated by Equation 10. By the calculations of this equation (Figure 2), the constancy of Neλ 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 ρck = 0. It follows that both of the trends pointed out by Balloux et al. (2003) and Bengtsson (2003) are true under Poisson-distributed (C = 1 and K = 1 +β) and independent (ρck = 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
