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Evolution of Genetic Variability and the Advantage of Sex and Recombination in Changing Environments
Reinhard Bürgeraa Institut für Mathematik, Universität Wien, A-1090 Wien, Austria and International Institute of Applied Systems Analysis, A-2361 Laxenburg, Austria
Corresponding author: Reinhard Bürger, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria., reinhard.buerger{at}univie.ac.at (E-mail)
Communicating editor: A. G. CLARK
 | ABSTRACT |
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TOP
ABSTRACT
THE GENERAL MODELThe role of recombination and sexual reproduction in enhancing adaptation and population persistence in temporally varying environments is investigated on the basis of a quantitative-genetic multilocus model. Populations are finite, subject to density-dependent regulation with a finite growth rate, diploid, and either asexual or randomly mating and sexual with or without recombination. A quantitative trait is determined by a finite number of loci at which mutation generates genetic variability. The trait is under stabilizing selection with an optimum that either changes at a constant rate in one direction, exhibits periodic cycling, or fluctuates randomly. It is shown by Monte Carlo simulations that if the directional-selection component prevails, then freely recombining populations gain a substantial evolutionary advantage over nonrecombining and asexual populations that goes far beyond that recognized in previous studies. The reason is that in such populations, the genetic variance can increase substantially and thus enhance the rate of adaptation. In nonrecombining and asexual populations, no or much less increase of variance occurs. It is explored by simulation and mathematical analysis when, why, and by how much genetic variance increases in response to environmental change. In particular, it is elucidated how this change in genetic variance depends on the reproductive system, the population size, and the selective regime, and what the consequences for population persistence are.
SINCE most higher organisms reproduce sexually, it may be argued that sexual reproduction and recombination are selectively favored over asexual reproduction. This selective advantage must be large because it has to overcome the twofold cost of sex that arises from the production of "needless" males. A gene that suppresses meiosis is transmitted to its offspring with certainty instead of a probability of one half and, therefore, should spread rapidly. A number of models and theories have been proposed to explain the selective maintenance of sexual reproduction and recombination (cf. 
The idea that sexual reproduction and recombination may be favored in changing environments has been revived relatively recently (
For various forms of environmental change, such as a directionally, periodically, or randomly changing optimum,
In a previous investigation of a quantitative-genetic model with a steadily moving optimum, 
500. This increase of variance had a positive effect on population persistence because it induced an accelerated response of the mean phenotype to the selective pressure.
More recently, KONDRASHOV and YAMPOLSKY (1996a,b) performed computer simulations of a diallelic multilocus model for a quantitative trait under a balance between mutation and fluctuating or periodic stabilizing selection. They reported huge increases of genetic variability, both for amphimictic and apomictic populations, and determined conditions under which amphimixis has an evolutionary advantage over apomixis. The finding of Kondrashov and Yampolsky that genetic variance may increase by a factor of 103 in amphimictic populations and somewhat less in apomictic populations is in sharp contrast to the results of
The purpose of this article is to investigate the evolutionary response of a population that has been under mutation-selection-drift balance to three kinds of environmental change: a directionally moving optimum, a periodically varying optimum, and a randomly fluctuating optimum. In particular, the evolutionary behavior of populations differing in their reproductive mode—sexual with recombination, sexual without recombination, and asexual—is studied. This is done by Monte Carlo simulations of a polygenic model in which each gene and each individual are represented and by mathematical analyses that use and extend work of
One of the main findings is that sexual populations differ significantly from asexuals in their ability to evolve genetic variance toward higher levels, as is favorable in many changing environments. In particular, under sexual reproduction and with recombination, the genetic variance under directional selection can increase on a short time scale and confer a substantial additional evolutionary advantage to recombination, because it leads to an accelerated response of the mean phenotype and, therefore, to a substantially reduced genetic load. Asexually reproducing populations, however, show no or almost no increase of genetic variance at least for the range of genetic parameters that is supported by experimental evidence. Thus, the advantage of sexual reproduction and recombination in coping with sustained, predictable environmental change may be much larger than previously thought. It is also investigated if the advantage caused by an increase in genetic variance, as it occurs in recombining populations, can be achieved by other means such as production of more offspring or a higher population size.
Concerning the evolution of genetic variation, this study yields results that are qualitatively different from those of KONDRASHOV and YAMPOLSKY (1996a). The reason for this difference is investigated and discussed. More generally, the genetic and demographic conditions are explored that lead to an increase of genetic variance in response to directional selection.
| THE GENERAL MODEL |
|---|
A finite population of diploid individuals that reproduces in discrete generations, either sexually with random mating and equivalent sexes or asexually, is considered. Fitness is determined by a single quantitative character under Gaussian stabilizing selection on viability, with the optimum phenotype 
t exhibiting temporal change. The viability of an individual with phenotypic value P is assumed to be
 |
(1) |
where 
2 is inversely proportional to the strength of stabilizing selection and independent of the generation number t. Selection acts only through viability selection, and each individual produces B offspring. Initial populations are assumed to be in a stationary state with respect to stabilizing selection and genetic mechanisms when environmental change commences. Three kinds of environmental change are considered:
- A phenotypic optimum that moves at a constant rate k per generation,
(2) A periodically fluctuating optimum,
(3) where A and L measure amplitude and period of the fluctuations, respectively.
- An optimum fluctuating randomly about its expected position, according to white noise with variance V and no autocorrelation.
Under each of these models, the population experiences a mixture of directional and stabilizing selection. Such models of selection were investigated previously by
The quantitative character under consideration is assumed to be determined by n mutationally equivalent loci that may be recombining or linked in the sexual case and are linked in the asexual case. The allelic effects are additive within and between loci; i.e., there is no dominance or epistasis. The phenotypic value of an individual is the sum of a genetic contribution and a normally distributed environmental effect with mean zero and variance 
2E = 1. Therefore, the phenotypic mean equals the mean of the additive genetic values, 
t, and the phenotypic variance is 2P,t = 2G,t + 2E, where 2G,t designates the additive genetic variance in generation t. The parameter Vs = 2 + 2E = 2 + 1 is used to describe the strength of stabilizing selection on the breeding values. It may be noted that Vs + 2G,t = 2 + 2P,t.
Because this article is concerned with finite populations of effective size Ne, theoretical predictions are needed for the distribution of the mean phenotype. Let
 |
(4) |
denote a measure for the strength of selection. Under the assumption of a Gaussian distribution of phenotypic values, the distribution of the mean phenotype t+1 in generation t + 1, conditional on t and t, is Gaussian. Its expectation is given by
 |
(5) |
and its variance by 
. From this, the following recursion equations for the expected mean and the expected variance of can be derived:
 |
(6a) |
 |
(6b) |
(cf.
 |
(7a) |
where 
2t = Vs + 2G,t + V[
t], and its (multiplicative) growth rate is
 |
(7b) |
(
Under prolonged environmental change, mean fitness may become very low. Because it is assumed that individuals can produce only a finite number, B, of offspring, a constant population size cannot necessarily be maintained. Therefore, a simple kind of density-dependent population regulation is imposed, which is described in the next section.
The simulation model and part of the theory are based on an explicit genetic model that requires specifying the mechanism by which genetic variability is maintained. It is assumed that this mechanism is mutation. Following CROW and KIMURA's (1964) continuum-of-alleles model, at each locus an effectively continuous distribution of possible effects for mutants is assumed. Thus, provided an allele with effect x gives rise to a mutation, the effect of the mutant is x + 
, where is drawn from a distribution with mean zero, variance 
2, and no skewness (for instance, a normal distribution). The mutation rate per haploid locus is denoted by µ, the genomic mutation rate by U, and the variance introduced by mutation per generation per zygote by Vm = U2.
| THE SIMULATION MODEL |
|---|
The analytical results in this article as well as in previous articles on related topics (
The simulation model has been adapted from the one used in 2 = 0.05 is assumed and a (diploid) genomic mutation rate of U = 0.02 per individual and generation. This implies that the variance introduced by mutation per generation is Vm = 0.001. These values have been suggested as gross averages by reviews of empirical data (cf. 2E = 1.
The generations are discrete, and the life cycle consists of three stages: (i) random sampling without replacement of a maximum of K reproducing adults from the surviving offspring of the preceding generation; (ii) production of offspring, including mutation and, possibly, segregation and recombination; and (iii) viability selection according to (1).
The maximum number K of reproducing adults may be called the carrying capacity. The Nt(
K) adults in generation t produce BNt offspring, an expected RtNt of which will survive viability selection. In this way, demographic stochasticity is induced. In a sexually reproducing population, the sex ratio is 1:1 and N/2 breeding pairs are formed, each producing exactly 2B offspring, while in the asexual case each adult produces B offspring. If the actual number of surviving offspring is >K, then K individuals are chosen randomly to constitute the next generation of parents. Otherwise, all surviving offspring serve as parents for the next generation. With this type of density-dependent population regulation, the number of reproducing adults remains roughly constant at K unless mean fitness decreases such that 
t < 
. If this occurs, the population cannot replace itself. The effective population size is Ne = 4N/Vf + 2), where Vf = 2(1 - 1/B)[1 - (2B - 1)/(BN - 1)] is the variance in family size. For further details see
For each parameter combination, a certain number of replicate runs with stochastically independent initial populations were performed. Each run was 105 generations unless population extinction occurred previously. Initial populations were obtained from a preceding initial phase of several thousand generations (depending on K) during which mutation-selection balance had been reached. The number of replicate runs per parameter combination was between 3 (for populations >2000 individuals and if no extinction occurred during the first 105 generations) and 100 (if extinction occurred in <1000 generations). Statistics were averaged over all generations of replicate runs.
| MUTATION-SELECTION-DRIFT BALANCE |
|---|
Understanding the evolutionary response of sexual and asexual populations in variable environments requires, as we shall see, a quantitative understanding of the stationary distribution under a balance between stabilizing selection, mutation, and random genetic drift. Models for the balance between mutation and stabilizing selection have been studied intensively. For fairly comprehensive reviews and recent results, refer to
For a single haploid locus and an infinitely large population, 
2(G) = 
, provided the variance of mutational effects, 2, is sufficiently small. A second-order approximation for the equilibrium variance was derived by 2/(µVs) << 1 (see below). Numerical simulations (2 < 9µVs. In this case, the equilibrium distribution has a low kurtosis and deviates only slightly from a Gaussian. By contrast, if 2 > 20µVs then TURELLI's (1984) house-of-cards (HC) approximation for the equilibrium variance, 2(HC) = 2µVs, is very accurate. In this case, the equilibrium distribution differs markedly from Gaussian. It has a high kurtosis and most variance is contributed by rare alleles of large effect.
In the present analysis, sexually reproducing diploid populations are compared with diploid asexuals. Both are assumed to have the same number of loci, n, and the same mutational parameters. The genome of an asexual diploid individual with n loci and no recombination can be identified with a single haploid locus subject to a mutation rate of µa = U = 2nµ. Following 2 < 9µaVs is satisfied if µa > 0.003. With the empirically suggested average value µa = 0.02, FLEMING's prediction will be accurate for asexuals and may be written as
 |
(8) |
where 
denotes the kurtosis of the mutation distribution ( = 0 for a normal distribution).
For sexually reproducing populations without recombination, the genetics is equivalent to that of a single diploid locus, and the equilibrium variance will be twice the haploid value but with µ = U/2, i.e.,
 |
(9) |
This is ~1.42a(F). Analogous arguments, but for the Gaussian prediction, may be found in
For freely recombining populations, the HC approximation applies to single loci, unless per-locus mutation rates are extremely high or stabilizing selection is extremely weak. If the n loci determining the trait are only loosely linked, then the approximation 2(HC) = 4nµVs = 2UVs for the additive genetic variance is valid (
So far, infinite population size has been assumed. For a finite population of effective size Ne, KIMURA's (1965) Gaussian approximation was extended to multiple loci by 2a, is approximately
 |
(10) |
For a nonrecombining sexually reproducing population of size Ne, the expected genetic variance 20 is approximately
 |
(11) |
Extensive computer simulations show that (10) and (11) provide excellent approximations to the observed average genetic variance and that the phenotypic distribution is nearly Gaussian unless Ne < 100 (results not shown).
For sexually reproducing populations and free recombination, the approximate formula
 |
(12) |
for the expected genetic variance was found independently and by different methods (
, (12) converges to the neutral formula 2(N) = 2NeVm of
From (7a), the expected mean fitness of a population at mutation-selection-drift balance is
 |
(13) |
where 2 = Vs(1 + 
) + 2G and 2G is the expected (additive) genetic variance at equilibrium. This produces a very accurate approximation to the mean fitness as calculated from computer simulations (results not shown).
Inspection of the formulas for the equilibrium genetic variance, (10), (11), and (12), shows that 2sex > 20 > 2a, provided all parameters are identical and 2UVs > 2. The ratio 
is approximately proportional to 
, thus weaker stabilizing selection leads to a larger genetic variance in freely recombining populations relative to nonrecombining and asexual populations. Because of the load induced by phenotypic variance, the mean fitness of freely recombining populations is always less than that of corresponding nonrecombining populations, however, typically only by a few percent. Moreover, asexual populations approach their infinite-size expectation at population sizes of a few hundred individuals, while sexual populations of the same size have variances considerably lower than their infinite-size expectation. This effect is most pronounced with moderate or weak stabilizing selection. Therefore, for populations of effective sizes of up to several thousand individuals, the advantage of asexual vs. sexual reproduction in stable environments is lower than predicted from load arguments for infinitely large populations.
| A DIRECTIONALLY MOVING OPTIMUM |
|---|
The purpose of this section is to explore the consequences of recombination and the mode of reproduction (sexual vs. asexual) on mean persistence time, mean fitness, and adaptive capability of populations experiencing sustained directional environmental change. A critical role is played by the genetic variance and its evolutionary response to selection. The influence of important genetic and environmental parameters is also investigated.
Classical quantitative genetics predicts that a population experiencing directional selection responds by shifting its mean phenotype according to Equation 5 and Equation 6aEquation 6b. If this selection is caused by a changing optimal phenotype, then the mean phenotype lags behind the optimum. For a model in which the optimum moves at a constant rate k per generation, according to (1) and (2), a critical rate of environmental change kc has been identified beyond which extinction is certain because the lag increases from generation to generation, thus decreasing the mean fitness of the population below < , the level at which the population starts to decline. With a smaller population size, genetic drift reduces the genetic variance, which leads to an even larger lag, to a further decrease of mean fitness, and to rapid extinction (
If the rate of environmental change is sufficiently low, then the mean lags behind the optimum but, after several generations, evolves parallel to it, on average according to E[t] kt - 
, where, as in (4), s = 
and 2move is the asymptotic genetic variance. The variance of the mean, caused by random genetic drift, is V[t] 
(
 |
(14) |
where 2 = Vs(1 + 
Ne) + 2move.
The critical rate of environmental change kc is defined as the value of k such that the population can just replace itself, i.e., such that BE[move] = 1. Unless population size is very small or stabilizing selection extremely weak, kc can be approximated by
 |
(15) |
(
The formulas for the expected lag, k/s, the expected mean fitness (14), and for kc are deceptively simple because the determinants of the genetic variance have not yet been elucidated. Actually, the genetic variance also evolves in response to environmental change and settles down to a stationary value, 2move. This value depends on the width of the fitness function, the number of loci, the effective population size, the mutation parameters, and on the rate of environmental change k. It is higher than the equilibrium value under a resting optimum, unless population size is small (2move leads to a higher mean fitness under a moving optimum, unless k is very small [k2 < 
; cf. CHARLESWORTH 1993b]. For any given k, however, an extremely large increase of genetic variance may cause mean fitness to decrease, because of the stabilizing component of selection (cf. also
Fig 1 demonstrates the advantage that sex and recombination may confer to sexually reproducing populations in a directionally changing environment. As a function of the rate of environmental change, k, it compares the mean extinction time, the mean fitness, and the genetic variance for two sexually reproducing populations, one with and one without recombination, and for two asexual populations, which differ by the number of offspring produced. The data points are obtained from simulations, and the lines are from theory as explained below. For all data shown, the carrying capacity is K = 211 = 2048. The number of offspring produced per adult is B = 5 for the two sexual and one asexual population, and B = 50 for the other asexual population. This value is used to show that for a large range of parameters, even a 10-fold advantage of asexual reproduction would not counterweigh the advantage of sex in this kind of changing environment.
|
Fig 1 (top) displays the mean time to extinction for the four types of populations. For the whole range of k values, a recombining sexual population persists indefinitely, i.e., for >105 generations when simulations were stopped. Additional simulations (not displayed) show that for k = 0.42 this population still persists for >105 generations, while for k = 0.48 and k = 0.72, its mean extinction times are 24,000 and 17 generations, respectively. A nonrecombining segregating population can permanently withstand only a much lower rate of environmental change. It goes extinct within some 1000 generations if k = 0.07 and within <100 generations if k 
0.12. The situation is even worse for asexual populations with the same B. They go extinct within <1000 generations if k 0.05. A 10 times larger B leads to a longer persistence time for a small range of k values, but does not help for large k. As soon as k 0.12, which corresponds to 10% of a phenotypic standard deviation, asexual populations become extinct within <100 generations, while equivalent freely recombining populations can easily persist through such a period of environmental change. For the recombining and nonrecombining sexual, and for the asexual populations with B = 50 and B = 5, approximate values for the critical rate of environmental change are kc = 0.69, 0.085, 0.080, and 0.050, respectively. They agree reasonably well with the simulation results.
Fig 1 (middle) displays the observed mean fitness together with theoretical approximations obtained from (14) by substituting the following values for the genetic variance: for the two asexual populations, the stochastic Fleming approximation (10) is used for all k. This provides an accurate approximation for the observed mean fitness. For the two segregating populations and k > 0, no good analytical approximations for the variance 2move are available. Substitution of the observed values of genetic variance into (14) yields precisely the observed mean fitness, thus supporting the Gaussian theory (deviations are on the order of 0.1%; results not shown). To demonstrate the extent to which the observed evolution of the genetic variance increases mean fitness, the mean fitness of equivalent sexual populations is plotted, but using the equilibrium genetic variance (dash-dotted lines). The results demonstrate that already in a slowly changing environment (k 0.05, which corresponds to a rate of environmental change of ~5% of a phenotypic standard deviation per generation), the mean fitness of asexual populations is reduced by almost a factor of five compared with an otherwise equivalent, but recombining, population. Thus, competition between such populations will almost certainly lead to the loss of the asexual population.
Fig 1 (bottom) shows that the genetic variance increases substantially for a freely recombining population (up to approximately fourfold), while for an asexual diploid population the variance is nearly independent of k (the maximum increase being 20%), and for the segregating but nonrecombining population the maximum increase is ~40%. The variance of the asexual population with B = 50 is almost identical (smaller by ~3%) to that of the population with B = 5, because its effective size is smaller by ~10%. For k = 0, all observed variances are very close to their analytical approximations with relative errors <5%.
The amount by which the genetic variance increases relative to its equilibrium value depends on population size. Because of the stabilizing component of selection, the variance must remain bounded even in an infinite population. However, it is not clear how large a population has to be to approach this upper limit. For a sexual and an asexual population, Fig 2 displays the ratio of the stationary variance under a moving optimum to that under a resting optimum as a function of effective population size. The lines on the right-hand side are for Ne = . Their values were determined by numerical iteration of the deterministic one-locus mutation-selection equation with a large number of alleles and extrapolation to 50 loci by assuming global linkage equilibrium (LE).
|
Information about linkage disequilibrium was obtained by calculating the LE variance, i.e., the variance computed from the gene frequencies by assuming Hardy-Weinberg proportions and LE [
Additional simulations were performed for extremely weak (Vs = 100) and very strong (Vs = 5) stabilizing selection. Under weak selection, recombining populations responded with a much smaller increase of genetic variance (
= 1.26 for k = 0.04 and Ne = 2276 compared with = 3.54 for Vs = 10), while under strong selection, the increase was even larger ( = 4.70, all other parameters unchanged). By contrast, asexual populations had their variance reduced under weak stabilizing selection ( = 0.62 for k = 0.012 and Ne = 2276), although theory predicts that an increase would enhance mean fitness. Under very strong selection and all other parameters unchanged, the variance of asexual populations also increased ( = 1.49). Because of their much greater genetic variance, the fitness advantage of sexual populations over asexuals even increased with weaker selection (data not shown). Given our knowledge about strength of stabilizing selection, values of Vs are more likely to be >10 instead of lower (
To explore the role of the mutation rate, simulations were performed with smaller mutation rates, i.e., with a genomic mutation rate of U = 0.002. With this mutation rate and all other parameters as in Fig 1, asexual populations responded to a moving optimum with an increase in variance of up to 80% (compared with 24% in the case U = 0.02), while freely recombining sexual populations experienced a maximum increase of variance by a factor of nine (compared with a factor of four if U = 0.02).
| A PERIODICALLY CHANGING OPTIMUM |
|---|
While the moving-optimum model may be applicable to sustained, long-term environmental changes, such as climatic trends, many species are subject to changes on a much shorter time scale. This is particularly true for species with short generation times. A frequently encountered scenario may be that of a periodically varying environment. We investigate population persistence, adaptive evolution, and the role of genetic variance in such an environment and compare recombining sexual populations with nonrecombining sexuals and diploid asexuals.
For the Gaussian phenotypic model, 
. Otherwise, it underestimates the true value substantially.
The subsequent theory follows the basic approach of Charlesworth but derives an approximation that is accurate for a wide range of parameters. It assumes a Gaussian distribution of phenotypic values. The starting point is Equation 6a, which allows representation of the mean phenotype in generation t, t, as a sum of sines, parallel to Equation 28 of
 |
(16) |
where st is the selection intensity (4). Defining the lag by 
t = t - t and averaging over one period of the cycle (after a sufficiently long initial phase has elapsed), one arrives at the approximation for the expected squared lag
 |
(17) |
where s is the asymptotic value of st The latter approximation in (17) requires (2
/L)2 << 1. Substitution of (17) into log 
t, as calculated from (7a), produces an accurate approximation for the expected log-mean fitness.
An approximation for the average mean fitness, per, can be obtained from (17) and (7a) by a straightforward Taylor approximation. Defining
we get
 |
(18) |
where 
From (18), sufficient (approximate) conditions can be derived ensuring that an increase in genetic variance causes an increase in mean fitness. These are
 |
(19) |
Fig 3 displays the mean fitness as function of the genetic variance for five different periods, L, and (top and bottom) for two different amplitudes. It shows that for small periods L, when the optimal phenotypes change by major jumps, genetic variance is detrimental to mean fitness, while for medium or long periods, more genetic variance is beneficial unless the variance is extremely large. This is similar to LANDE and SHANNON's (1996) results but in partial contrast to
|
The detailed evolution of finite sexual and asexual populations in a periodically changing environment was investigated by Monte Carlo simulations as described above. No assumptions are imposed on the distribution of phenotypic (or genotypic) values. Some of the results are summarized in Fig 4 and Fig 5. The quantities of interest are displayed as functions of k = 4A/L, which can be interpreted as the rate of change of the optimum averaged about one full cycle (during which the optimum moves 4A units, measured in multiples of 2E). Dynamically, an infinite period L is equivalent to a resting optimum.
|
|
In Fig 4, the amplitude A of the periodic optimum is chosen to be 2, which implies that, at the most extreme position of the optimum (A units from the origin), the originally optimal phenotype (at position 0) has a fitness of 13.5%. The top shows that in this case, only the freely recombining population persists for all possible periods, while the nonrecombining sexual and the asexual die out for intermediate periods L. The middle shows that the freely recombining population has a much higher mean fitness than the two other populations if the length of the period is between 100 and 500. The lines in the middle represent the mean fitness that would be obtained if the variance did not increase under the periodic optimum. This yields good approximations for the nonrecombining and the asexual population, but not for the recombining population, because its genetic variance increases substantially unless the period, L, is very short (L 20). This is shown in the bottom. Thus, for this kind of environmental change, recombination improves mean fitness and increases population persistence only if the period of the cycle is long, i.e., evolution is slow. This advantage of recombination is due to the fact that it enables populations to increase in genetic variance. If the optimum changes rapidly, more genetic variance is not beneficial. In this case, it makes sense for a population to stay where it is and wait until the environmental optimum returns. Clearly, this requires that the population is able to maintain a minimum viable population size during the period of low fitness.
Fig 5 is similar to Fig 4, but with a higher amplitude of environmental fluctuations (A = 3). Then the fitness of the originally optimal phenotype is only 1.1% if the environmental optimum is at its most extreme position, A. The figure compares a freely recombining sexual population with three different diploid asexuals: one with all other parameters equal, a second that produces 10 times as many offspring (B = 50), and a third whose carrying capacity is 8 times higher (K = 214, Ne = 18,205). Again, for long periods L, i.e., slow environmental change, the sexual population has the highest mean extinction time and mean fitness. For a rapidly changing optimum, however, it is the asexual population that produces B = 50 offspring that has the longest persistence time, although it has a much lower genetic variance than the sexual population (bottom). The lines (middle) display the mean fitness as expected with a constant genetic variance. This figure clearly demonstrates that in a rapidly changing, periodic environment, it is not the flexibility of the genome and the resulting better adaptability that improves population persistence. Instead, other strategies, such as production of more offspring, may be more successful. For a slowly changing optimum, good adaptation is the best response to environmental change.
Further simulations (results not shown) have been performed with smaller population sizes and lower amplitudes (A = ). Although for lower population sizes the increase of variance in freely recombining populations is generally much lower, they still persist longer and have a higher mean fitness than otherwise equivalent but nonrecombining sexual or asexual populations. Again, the advantage of sex and recombination is most pronounced for large amplitudes A and long periods L.
If mutational effects are drawn from a reflected gamma distribution instead of a Gaussian (cf.
Finally, it is worth mentioning that if the observed genetic variance is substituted into Equation 18, a very accurate approximation of the observed mean fitness is obtained, thus supporting the mathematical theory.
| A RANDOMLY FLUCTUATING OPTIMUM |
|---|
The Gaussian theory of , and no autocorrelation, an increase of genetic variance leads to an increase of mean fitness only if V > 2Vs. A similar result was obtained by
Monte Carlo simulations with the model described above have shown that in both asexual and sexual populations the genetic variance remains virtually constant if V Vs, in accordance with similar results of
In contradistinction to theory ( = Vs = 10, sexual freely recombining and asexual populations of effective size Ne = 1140 and with B = 5 went extinct after an average of ~7900 and 6600 generations, respectively (100 replicated runs). Asexual populations with twice the mutation rate and intermediate variance had a mean extinction time of 6900 generations. The mean fitnesses averaged over the persistence time, however, were ranked inversely to the genetic variances, but the differences were not significant. More genetic variance is beneficial in this case, probably because under rare large excursions of the optimum more individuals survive in populations with a broader distribution. Hence, the theory based on the assumption of a Gaussian phenotypic distribution predicts an advantage for lower genetic variability and asexual reproduction when there is none. If, however, asexuals have twice the growth rate of sexuals, then their mean persistence time is approximately tripled and, thus, higher than that of the corresponding sexual population.
For smaller V, asexual populations have a slightly higher mean fitness than sexuals because of their lower genetic variance (results not shown). Indeed, monomorphic populations have the highest fitness and extinction time under such a model (cf.
| DISCUSSION |
|---|
This investigation departs from previous work on the advantage of sex and recombination in changing environments (cited in the Introduction) primarily by using an explicit genetic multilocus model and by assuming finite population size and population regulation. Genetic variation is assumed to be maintained by mutation-selection balance. In a constant environment that exerts stabilizing selection on the trait under consideration, sexually reproducing populations have more genetic variance than mutationally and demographically equivalent asexual populations. This leads to a slightly higher equilibrium load of sexual populations. If, however, adaptation becomes necessary due to environmental change, then sexual populations can respond faster and gain a substantial selective advantage over asexuals.
In accordance with results of
In a constant environment, in a randomly fluctuating environment with small variance and no autocorrelation, and in a rapidly changing periodic environment with short periods, asexually reproducing populations have a higher mean fitness and a longer persistence time than freely recombining sexual populations. This is in qualitative agreement with the Gaussian theory (