Genetics, Vol. 166, 1529-1539, March 2004, Copyright © 2004

The Effective Size of Mixed Sexually and Asexually Reproducing Populations

Katsuei Yonezawaa, Takuro Ishiib, and Tsukasa Nagamineb
a Department of Biotechnology, Kyoto Sangyo University, Kyoto 603-8555, Japan
b Research Team of Plant Resources, National Institute of Agrobiological Sciences, Tsukuba 305-8602, Japan

Corresponding author: Katsuei Yonezawa, Kyoto Sangyo University, Kamigamo, Kita-ku, Kyoto 603-8555, Japan., yonezaw{at}cc.kyoto-su.ac.jp (E-mail)

Communicating editor: T. H. D. BROWN


*  ABSTRACT
*TOP
 *ABSTRACT
*FORMULATIONS OF THE EFFECTIVE...
 *INFLUENCE OF ASEXUAL...
 *DISCUSSION
 *LITERATURE CITED

Using the transition matrix of inbreeding and coancestry coefficients, the inbreeding (NeI), variance (NeV), and asymptotic (Ne{lambda}) effective sizes of mixed sexual and asexual populations are formulated in terms of asexuality rate ({delta}), variance of asexual (C) and sexual (K) reproductive contributions of individuals, correlation between asexual and sexual contributions ({rho}ck), selfing rate (ß), and census population size (N). The trajectory of NeI toward Ne{lambda} changes crucially depending on {delta}, N, and ß, whereas that of NeV is rather consistent. With increasing asexuality, Ne{lambda} either increases or decreases depending on C, K, and {rho}ck. The parameter space in which a partially asexual population has a larger Ne{lambda} than a fully sexual population is delineated. This structure is destroyed when N(1 – {delta}) < 1 or {delta} > 1 – 1/N. With such a high asexuality, tremendously many generations are required for the asymptotic size Ne{lambda} to be established, and Ne{lambda} is extremely large with any value of C, K, and {rho}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 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 Down; KAWANO 1984 Down; RICHARD 1986 Down; SMITH and SZATHMARY 1999 Down). 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 Down, WRIGHT 1969 Down 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 Down; HARTL and CLARK 1989 Down; CABALLERO 1994 Down; WANG and CABALLERO 1999 Down) as well as in solving various optimization issues in population management (FRANKLIN 1980 Down; SOULE 1980 Down; LANDE 1995 Down; SANTIAGO and CABALLERO 1995 Down). Various kinds of Ne have been defined, such as for inbreeding, variance, and eigenvalue, denoted here NeI, NeV, and Ne{lambda}, respectively (EWENS 1982 Down; CROW and DENNISTON 1988 Down). 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 Ne{lambda} with increasing generations in sexually (but not fully selfing) reproducing populations (EWENS 1979 Down; POLLAK 1987 Down; CHESSER et al. 1993 Down). 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 Down and BALLOUX et al. 2003 Down 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 Down; YONEZAWA et al. 2000 Down), 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
*TOP
 *ABSTRACT
 *FORMULATIONS OF THE EFFECTIVE...
*INFLUENCE OF ASEXUAL...
 *DISCUSSION
 *LITERATURE CITED

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 ({delta}), 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 ß. 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

(1)

where {theta}t–1 is the coancestry of generation t – 1 and {delta} equals the mean asexuality rate of individuals.

The coancestry {theta}t is obtained as a sum of three components {theta}1t, {theta}2t, and {theta}3t, 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 {theta}1t is given by

(2)

where ci is the number of asexual progeny of individual i of generation t – 1, and N{delta} is the total number of asexual individuals in the population. Approximating N – 1 to N and using symbols W1 and W2, defined as

Equation 2 can be expressed simply as

(3)

Variables and Vc in the equation of W1 stand for the mean and variance of the asexual progeny per individual, respectively. is equal to {delta} under a constant population size. The standardized variance Vc/, denoted C hereafter, is equal to 1 when the asexual reproductive contribution of the individuals (ci) is Poisson distributed, being larger or smaller than 1 depending on whether ci is more or less dispersed than Poisson distributed. The second component {theta}2t is given by

(4)

where si is the number of progeny of individual i produced by selfing, and k'i is the number of gametes (either female or male) of individual i transmitted through outcrossing. The sum 2si + k'i, denoted ki, indicates the total number of gametes contributed by individual i. Under a constant population size, si, k'i, and ki have means (1 – {delta})ß, 2(1 – {delta})(1 – ß), and 2(1 – {delta}), respectively. The mean and variance of ki are denoted by and Vk, respectively. The standardized variance Vk/, denoted K, is equal to, larger than, or smaller than (1 + ß) depending on whether the sexual progeny size (si + k'i) per individual is Poisson distributed, more dispersed, or less dispersed than Poisson distributed, respectively. Using symbols X1 and X2, defined as

Equation 4 is written as

(5)

Symbol {rho}ck 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 Down; KAWANO 1984 Down; WALLER 1988 Down) and was newly incorporated in this article.

The third component is presented as

(6)

which can be simplified to

(7)

where

From Equation 1, Equation 3, Equation 5, and Equation 7, the transition matrix of inbreeding is obtained as

(8)

where T11 = {delta} +(1 – {delta})ß/2, T12 = (1 – {delta})(1 – ß), T21 = (W1 + X1 + Y1)/2, T22 = (W2 – W1) + (X2 – X1) + (Y2 – Y1), U1 = (1 – {delta})ß/2, and U2 = T21. Using the relations ft = 1/(2NeI,t) + {1 – 1/(2NeI,t)}ft–1 and {theta}t = 1/(2Ne{theta},t) + {1 – 1/(2Ne{theta},t)}{theta}t–1, the instantaneous effective sizes for inbreeding and coancestry, denoted NeI,t and Ne{theta},t, are obtained by (1 – ft–1)/{2(ft – ft–1)} and (1 – {theta}t–1)/{2({theta}t – {theta}t–1)}, respectively. Ne{theta},t determines the progress of genetic fixation due to random drift (COCKERHAM 1973 Down; POLLAK 1987 Down) and can be treated as the variance effective size, denoted NeV,t. As generations advance, both NeI,t and NeV,t asymptote to the same value, 1/{2(1 – {lambda}1)} (EWENS 1979 Down), where {lambda}1 is the first eigenvalue of the matrix [Tij] of Equation 8, being formulated as

(9)

Neglecting the second- or higher-order terms of 1/N, Ne{lambda} is derived as

(10)

In the case when the sexual and asexual reproductive contributions are independent ({rho}ck = 0), Equation 10, as it should, becomes the same as Equation 5 of YONEZAWA 1997 Down, which was derived by ignoring the reproductive correlation [note that the asymptotic value of the deviation from the Hardy-Weinberg proportions, denoted {alpha} previously, is equal to ß/(2 – ß) not only in a fully sexually reproducing population (HALDANE 1924 Down) but also in a partially asexually reproducing population (MARSHALL and WEIR 1979 Down)]. With {delta} = 0, Equation 10 equals Equation 8 of WANG 1996 Down under constant population size and selfing rate.


*  INFLUENCE OF ASEXUAL REPRODUCTION ON THE EFFECTIVE POPULATION SIZE
*TOP
 *ABSTRACT
 *FORMULATIONS OF THE EFFECTIVE...
 *INFLUENCE OF ASEXUAL...
*DISCUSSION
 *LITERATURE CITED

Instantaneous effective size:
NeI,t and NeV,t, under a Poisson-distributed asexual and sexual reproductive contribution (C = 1 and K = 1 + ß) with the initial condition f0 = {theta}0 = 0, were calculated for some typical values of the parameters involved. As illustrated in Fig 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 – {delta})ß} from the transition matrix (8), takes quite different values depending on the values of {delta} and ß.



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  		Figure 1. 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 ({delta}) and selfing (ß), calculated with specifications N = 100, f0 = {theta}0 = 0, {rho}ck = 0, C = 1, and K = 1 + ß. The half-life generations tI,0.5 and tV,0.5 were calculated by tI,0.5 = 1 + log Q/log({lambda}2/{lambda}1), where , and {lambda}2 equals the second eigenvalue of the matrix [Tij], and tV,0.5 = 1 + log Q'/log({lambda}2/{lambda}1), where , and .

As generations proceed, NeI,t asymptotes either downward or upward to the eigenvalue effective size Ne{lambda}, depending on whether NeI,1 is larger or smaller than Ne{lambda}. NeI,t is constant over generations when NeI,1equals Ne{lambda}. A critical value of {delta}, denoted , can be obtained by solving the equation NeI,1 = Ne{lambda} such that the trajectory either declines or rises depending on whether {delta} is larger or smaller than . Under a Poisson-distributed reproductive contribution with independent asexual and sexual reproduction (C = 1, K = 1 + ß, and {rho}ck = 0), is derived as

(11)

which takes a larger value with a larger population size and/or selfing rate. In a nearly fully outcrossing (ß {cong} 0) or random mating (ß = 1/N) population, takes a negative value, indicating that, as seen from Fig 1, NeI,t asymptotes downward to Ne{lambda} at any value of {delta}. By contrast, is ~1 – 2/N when ß is close to 1, meaning that NeI,t asymptotes upward to Ne{lambda} at practically any {delta} unless N is relatively small. The calculations under ß = 0.1 and N = 100 in Fig 1 show that the asymptotic direction of NeI,t is reversed at a certain asexuality rate between 0.5 and 0.95, which is actually 0.894. As expected, NeI,t approaches Ne{lambda} more slowly at a higher asexuality. To indicate the rate (quickness) of the asymptotic approach, the half-life generation, denoted tI,0.5, at which the initial (t = 1) deviation of NeI,t from Ne{lambda} is halved, is presented in Fig 1 (see the Fig 1 legend for the calculation of half-life generation). NeV,t asymptotes in a much smaller magnitude and simpler pattern than NeI,t. As can be derived from Equation 8 and Equation 10, a relation Ne{lambda} = (1 – ß/2)NeV,1 holds with any values of C, K, and {rho}ck. Therefore, as shown in Fig 1, NeV,t is almost constant in a mainly outcrossing population (ß < 0.1) while it decreases by 50% in a fully selfing population. When {delta} is large and ß is small, NeV,t decreases quite slowly and its half-life generation, denoted tV,0.5, is larger than tI,0.5. The correlation {rho}ck does not produce any important change in the dynamic patterns of NeI,t and NeV,t, although the trajectories are shifted upward when {rho}ck < 0 and downward when {rho}ck > 0 (data not presented).

Asymptotic effective size:
It is seen from Equation 10 that the asymptotic effective size Ne{lambda} is the same at any rate of asexuality when C = (K + 1 – ß)/2 and {rho}ck = 0. The condition C = (K + 1 – ß)/2 holds under Poisson-distributed reproductive contribution (C = 1 and K = 1 + ß). Otherwise Ne{lambda} changes with increasing asexuality rates in different patterns depending on the values of C, K, and {rho}ck (Fig 2). When {rho}ck = 0, the relative magnitude of C to (K + 1 – ß)/2 alone counts; Ne{lambda} increases with increasing {delta} when C < (K + 1 – ß)/2 (case 2 in Fig 2), whereas it decreases when C > (K + 1 – ß)/2 (case 3). This trend is considerably modified in the presence of a reproductive correlation; {rho}ck acts to enlarge Ne{lambda} when negative and to decrease it when positive.



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  		Figure 2. Effect of asexuality rate on Ne{lambda} under different patterns of reproductive contribution of individuals, calculated with ß = 0.

The effect and interrelationship of the various reproductive parameters can be more comprehensively grasped on the basis of the term {delta}(2C – K – 1 + ß) + 2{rho}ck , a component of the denominator of Equation 10. Ne{lambda} of a partially asexual population is larger than, equal to, or smaller than that of a fully sexual population, depending on whether this term is negative, zero, or positive; in other words, C is smaller than, equal to, or larger than a critical value defined as

(12)

which is the same as that obtained previously (YONEZAWA 1997 Down) under {rho}ck = 0. C0 increases when {rho}ck is negative, and then the condition for asexuality to be advantageous (C < C0) becomes less severe. The condition C < C0 is met at any asexuality rate when {rho}ck < 0 and C <= (K + 1 – ß)/2. To visualize the parameter space that satisfies C < C0, line C = C0 was drawn in Fig 3 for some typical combinations of K, ß, and {rho}ck. In this figure, three regions, i.e., above, on, and below the line C = C0 (named phases I, II, and III in YONEZAWA 1997), are delineated; a population that locates above, on, or below the line has a Ne{lambda} that is smaller than, equal to, or larger than that of a fully sexual population ({delta} = 0). A negative reproductive correlation acts to raise the slope of the line and therefore to widen the region of phase III, whereas a positive correlation produces the opposite effect. This effect of {rho}ck is more prominent with a lower rate of asexuality. Selfing acts to shift down the line, narrowing the region of phase III.



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  		Figure 3. Delineation of the condition for asexuality causing a positive or negative effect on the effective size (Ne{lambda}). The intercept of the line is (1 – ß)/2. See text for the interpretation of this figure.

Highly asexual and small populations:
When the terms of the second as well as the first order of 1/N are considered, Ne{lambda} is formulated as

(13)

where D stands for the denominator of Equation 10. Equation 13 shows that Ne{lambda} should be larger than that obtained by Equation 10 when the number of sexual individuals, N(1 – {delta}), is <1; in other words, the asexuality rate is >1 – 1/N. Therefore, when calculated with a small, highly asexual population, Ne{lambda}/N could be significantly larger than that calculated by Equation 10. The calculations with population sizes 50 and 200 (Fig 4) give the same trend pointed out by BALLOUX et al. 2003 Down that Ne{lambda} increases suddenly toward infinity when asexuality becomes complete. The sudden increase in Ne{lambda} occurs with any values of the reproductive parameters C, K, and {rho}ck. With the same asexuality rate, the increase is more prominent in a smaller population.



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  		Figure 4. Effect of asexuality rate on Ne{lambda} when calculated by the exact eigenvalue effective size. Calculated with ß = 0 and {rho}ck = 0.

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 Down; WANG and CABALLERO 1999 Down), and then the effective to census size ratio Ne{lambda}/N is constant over changing N. It is known by Equation 13 that the constancy of Ne{lambda}/N is destroyed when {delta} > 1 – 1/N. The calculations of Fig 5 show that with an asexuality rate as high as 0.99, Ne{lambda}/N increases markedly with decreasing N. It may be said that asexuality has a buffering effect of stabilizing Ne{lambda} against decreasing N.



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  		Figure 5. Ratio of the effective to census population size (Ne{lambda}/N) under varying census population sizes (N). Ne{lambda} was calculated using 1/{2(1 – {lambda}1)} with specifications ß = 0, C = K = 1, and {rho}ck = 0.

The mechanism of the above-mentioned effect of asexuality can be explicitly interpreted when considering the dynamics of Ne of a fully asexual population ({delta} = 1). Inbreeding does not progress under complete asexuality, and therefore NeI,t is always infinite. With f0 = {theta}0 = 0, NeV,t is derived from Equation 8 as

(14)

where B = 1 – C/N. As it should, NeV,t is also infinite when each of the N individuals leaves one progeny (C = 0). In a sufficiently large population satisfying C/N {cong} 0, NeV,t is almost constant and equal to N/C. With {delta} and ß being substituted by 1 and 0, respectively, Equation 10 gives the same effective size N/C, coinciding with the prediction of EWENS 1982 Down that NeV and Ne{lambda} are usually close when they exist. The constancy over generations of NeV,t, however, is destroyed in the long run unless N is infinitely large; NeV,t increases consistently toward infinity as generations advance, reflecting that while genotypic diversity in the population is lost gradually due to random genetic drift, allelic diversity within individuals is kept unchanged. Eventually, the population is totally occupied by copies of an ancestral genotype and allelic diversity within this genotype never decays. At this state, the coancestry {theta}t is fixed to 1/2 [(1 + f0)/2 if the inbreeding coefficient is f0 initially], and, by definition, NeV,t should be infinite. This state is established more rapidly with smaller populations (smaller B).

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 Ne{lambda} (Fig 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 Ne{lambda} 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 (Fig 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 , where {Delta}{theta}i is the increment of the coancestry between generations i – 1 and i and NeVh is the harmonic mean of instantaneous variance effective sizes of the t generations, NeVh is obtained as

(15)



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  		Figure 6. Asymptotic patterns of inbreeding and variance effective sizes under extremely high rates of asexuality, calculated with N = 100, f0 = {theta}0 = 0, ß = 0, C = K = 1, and {rho}ck = 0.

Similarly, the harmonic mean of instantaneous inbreeding effective sizes is obtained as

(16)

When the reproductive parameters such as C, K, ß, {rho}ck, and {delta} are estimated for the population concerned, NeVh and NeIh can be readily calculated using the transition matrix defined before. The initial inbreeding coefficient f0 for this calculation should be estimated by the deviation from the Hardy-Weinberg proportions, and the initial coancestry {theta}0 should be zero.

The variance harmonic mean NeVh is stable over many generations; e.g., in Fig 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

(17)



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  		Figure 7. Harmonic mean over t generations of the instantaneous inbreeding and variance effective sizes under extremely high rates of asexuality, calculated with N = 500, f0 = {theta}0 = 0, C = 1, K = 1 + ß, and {rho}ck = 0.

When the population initially has an asymptotic genetic structure f0 = ß/(2 – ß), NeV,1, as it should, is immediately equal to the asymptotic effective size Ne{lambda} defined in Equation 10. When heterozygotes dominate, which might occur in a population as asexual as N(1 – {delta}) < 1, f0 takes some negative value and then NeV,1 could be very large.

In contrast to variance effective size, inbreeding effective size decreases toward Ne{lambda} as generations proceed (Fig 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 (Fig 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 ({rho}ck < 0), asexuality could be highly advantageous to retain genetic diversity (Fig 2). In plant species, highly asexual populations very often show poorer genetic diversities than sexual populations (PLEASANTS and WENDEL 1989 Down; ASPINWALL and CHRISTIAN 1992 Down; PELLEGRIN and HAUBER 1999 Down). 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 (BALLOUX et al. 2003 Down). 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
*TOP
 *ABSTRACT
 *FORMULATIONS OF THE EFFECTIVE...
 *INFLUENCE OF ASEXUAL...
 *DISCUSSION
*LITERATURE CITED

Extension to generation-overlapping populations:
Many of the partially or fully asexually reproducing plant species are perennial (HARPER 1977 Down; KAWANO 1984 Down; WALLER 1988 Down). 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 {theta}t is formulated in a form like Equation 8. Then, similar mathematical procedures as used above give the annual effective sizes (HILL 1972 Down) for ft and {theta}t, denoted NyI,t and NyV,t, respectively, both of which asymptote to a common asymptotic annual effective size Ny{lambda}. 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 Down). Because a relation

holds, NeI,T is derived as

which equals a weighted harmonic mean of the annual effective sizes over [L] years. After NyI,t has reached the asymptotic size Ny{lambda}, NeI,T becomes

(18)

where [L] is the mean inbreeding coefficient over [L] years, i.e., . NeI,T asymptotes to Ny{lambda}/[L] as generations proceed. The lifetime variance effective size NeV,T is derived in the same way, using the relation NeV,T = (1 – {theta}T–1)/{2({theta}T – {theta}T–1)}.

Conservation genetic interpretation:
The reproductive pattern (C, K, and {rho}ck) 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 ß/(2 – ß) 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 {rho}ck being defined as presented in Table 1.


 
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  		Table 1. Effective population sizes under three typical schemes of controlled propagation

The calculations illustrated in Fig 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 ß gets close to unity; Ne of all three equals uN/(1 – u) when ß = 1.



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  		Figure 8. Ne under the three propagation schemes I, II, and III in response to varying sampling fractions (u).

Coalescence effective size:
ORIVE 1993 Down and BALLOUX et al. 2003 Down 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 {rho}ck are difficult to incorporate explicitly and instantaneous Ne cannot be known by the coalescence time theory.

BALLOUX et al. 2003 Down 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, Ne{lambda} either increases or decreases with increasing asexuality, depending on the reproductive parameters C, K, and {rho}ck (Fig 2 and Fig 4). The trend pointed out by BALLOUX et al. 2003 Down occurs when C = 1, K = 1 + ß, and {rho}ck = 0.

The calculations of coalescence time of BENGTSSON 2003 Down 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 Fig 1) are produced per generation. In the context of our approach, his finding can be expressed as asexuality ({delta}) causes no significant change in Ne{lambda} when the number of sexual individuals N(1 – {delta}) is larger than a few. The term of the second order of 1/N in our Equation 13 can be ignored when N(1 – {delta}) is larger than a few, and then Ne{lambda} can be approximated by Equation 10. By the calculations of this equation (Fig 2), the constancy of Ne{lambda} over changing {delta} and therefore the finding of BENGTSSON 2003 Down hold when C = 1, K = 1 + ß [more generally, C = (K + 1 – ß)/2], and {rho}ck = 0. It follows that both of the trends pointed out by BALLOUX et al. 2003 Down and BENGTSSON 2003 Down are true under Poisson-distributed (C = 1 and K = 1 + ß) and independent ({rho}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 Down). 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.

Manuscript received June 16, 2003; Accepted for publication December 10, 2003.


*  LITERATURE CITED
*TOP
 *ABSTRACT
 *FORMULATIONS OF THE EFFECTIVE...
 *INFLUENCE OF ASEXUAL...
 *DISCUSSION
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