Abstract
In this study we consider a mathematical model of a sexual population that lives in a changing environment. We find that a low rate of environmental change can produce a very large increase in genetic variability. This may help to explain the high levels of heritability observed in many natural populations. We also study asexuality and find that a modest rate of environmental change can be very damaging to an asexual population, while leaving a sexual population virtually unscathed. Furthermore, in a changing environment, the advantages of sexuality over asexuality can be much greater than suggested by most previous studies. Our analysis applies in the case of very large populations, where stochastic forces may be neglected.
ALL environments change with time. Understanding how populations respond to change is fundamental for evolutionary biology, agriculture, and conservation. However, this problem has proved resistant to analysis. A truly realistic model must allow for the occurrence of mutations with a wide variety of effects, and it must also allow for large population sizes. Unfortunately, including these factors in a mathematical model introduces profound difficulties into the analysis.
One way that researchers have dealt with the mathematical difficulties is to use computer simulations. Unfortunately, because of the limitations of computer memory and CPU time, computer analysis does not allow consideration of very large populations. As an example, Bürger and Lynch (1995) used computer simulations to study evolutionary dynamics in a changing environment. These authors allowed for genetic mutations with a wide range of effects. However, the largest population considered had only 512 members. In contrast, many natural populations have >10^{9} members. In addition, the perallele mutation rate used by Bürger and Lynch (2 × 10^{4}) was far higher than the rates usually thought to be biologically realistic (Griffithset al. 1996). The high mutation rate may also have been necessitated by computational constraints, as equilibration can be very slow when allelic mutation rates are low.
Many natural populations are very large. With this in mind, some investigators have focused on infinite populations, which should provide very similar results to finite populations of sufficient size. One example is a study by Charlesworth that considered the fate of infinite sexual and asexual populations in a changing environment (Charlesworth 1993). However, to carry out his study, Charlesworth made use of the infinitesimal model (Bulmer 1980). This model assumes that an infinite number of loci control the selected trait and that alleles at each locus have infinitesimal effects. These assumptions are clearly unrealistic, although the analysis does provide some important insights.
In this study, we attempt to make progress toward the formulation of a biologically realistic model of adaptation in a changing environment. Like Charlesworth, we assume that the population is infinite, and we consider both sexual and asexual populations. However, we present a new analysis that allows for consideration of the more realistic situation where a relatively small number of loci control the selected trait, and we also allow for the occurrence of mutations that have substantial effects. We find that the results from this analysis are very different from those of previous studies that have considered small populations and those of previous studies that have used less realistic genetic systems.
MODEL AND RESULTS
Evolution in an unchanging environment: Consider a population of obligately sexual hermaphrodites that is sufficiently large that stochastic effects can be ignored (i.e., the population is effectively infinite). Each individual is characterized by the value of a phenotypic trait such as height or weight. An individual’s measurement on the trait is denoted by z. We assume that there is an optimal value of z, called z_{opt}. The value of z_{opt} does not change over time, and death rate increases with the deviation of z from z_{opt} (thus we have stabilizing selection). In particular, for an individual with phenotypic value z, the probability of dying over a very small period of time, Δt, is given by D · Δt. Here, D represents death rate, and it is given by D = [1 + (z  z_{opt})^{2}/(2V)], where V > 0. The value of V is inversely related to the strength of stabilizing selection.
Extensive numerical study shows that, after a sufficiently long period of time has elapsed, the mean value of D tends to settle down to a particular value, which we denote by D_{S} (“S” stands for sexual). Note that for an individual with the optimal phenotype (i.e., z = z_{opt}), we have D = 1. Thus, in a typical population, where most individuals do not have the optimal phenotype, we have D_{S} > 1.
Individuals can produce offspring either by “male effort” (i.e., by providing sperm or pollen) or by “female effort,” which involves providing some of the resources necessary for maturation (as in seed production). At any given time, all individuals produce offspring via female effort at the same rate, B. Thus, during a very small period of time, Δt, the probability that a given individual will produce an offspring via female effort is given by B · Δt. Mating is random, and offspring mature instantaneously. We assume that, at any given instant, births compensate for deaths, so that population density does not change with time. Thus, after equilibration of death rate, we have B = D_{S}.
The phenotype of a particular individual is assumed to depend on its “genotypic value,” G, plus a normally distributed environmental noise component, ε (thus, z = G + ε). The distribution of ε is independent of G and has mean zero and standard deviation V_{e}. Without loss of generality, other variables are scaled so that V_{e} = 1. Furthermore, we assume that individuals are diploid with L freely recombining loci determining the value of G. Thus, in any individual, there are 2L locations within the genome where alleles that influence the value of G occur. These locations are numbered as i = 1, 2,..., 2L. The DNA sequence of the allele at location i determines its effect, x_{i}, on the phenotypic character, and ∞ < x_{i} < ∞. The allelic effects combine additively, and thus
Each of an individual’s 2L alleles is a copy of an allele present in one or the other of the individual’s parents. The value of x_{i} associated with a particular allele is identical to that of the parental allele, unless a mutation occurs during the copying process. The perallele rate of such mutations is denoted by μ (where 0 ≤ μ ≤ 1). We make the usual assumptions that mutations to different alleles occur independently and that mutant values of x_{i} are normally distributed around the parental value (Kimura 1965; Lande 1975; Turelli 1984). When a mutation occurs that alters the allele at location i, the probability that the mutant allele has an effect in the infinitesimal interval y + dy > x_{i} > y is given by f (y  x^{*}) dy. Here, x^{*} is the parental value of x_{i}. f (y  x^{*}) is given by
What parameter values are reasonable to use for the production of results from the model? At present there is a great deal of debate about this matter. Furthermore, the most realistic choice of parameter values probably depends on which phenotypic trait is under consideration. With this in mind, we produced results for four different sets of parameter values.
Experiments suggest that allelic mutation rates in multicellular organisms are generally on the order of 10^{5} (Griffithset al. 1996). Furthermore, most of the heritable variation for many easytomeasure traits (e.g., body size) appears to be controlled by <20 loci (Turelli 1984; Bürgeret al. 1989; Bürger and Lynch 1995; Kearsey and Farquhar 1998). Finally, rough estimates from the data (Turelli 1984; Lynch and Walsh 1998) suggest that, typically, m^{2} ⪡ 1, and V ≈ 20 [m ≈ 0.2 is often used in published calculations (Turelli 1984; Bürger and Lynch 1995; Lynchet al. 1995; Lynch and Walsh 1998)].
The foregoing considerations motivate the calculations made for Table 1, for which we set L = 10, μ = 10^{5}, m = 0.2, and V = 20. The first row of Table 1 gives the data for an unchanging environment. In this case, the genetic variance within a sexual population, V_{G}_{,S} is quite low (V_{G}_{,S} = 0.00807). (The genetic variance is defined as the variance in G values within the population, and the subscript S indicates that the population is sexual.) As a result of the low value of V_{G}_{,S}, heritability is also low. As noted previously, the environmental variance is scaled to unity, and thus the heritability (
The parameter values used for Table 1 imply that a very small amount of genetic variance results from newly arisen mutations. This “mutational variance” is equal to 2Lμm^{2}, and for Table 1, 2Lμm^{2} = 8 × 10^{6}. A more realistic number may be closer to 10^{3} (Turelli 1984; Bulmer 1989; Lynch and Walsh 1998). Methods used to calculate the number of loci involved in the control of a phenotypic trait may produce estimates that are much lower than the actual number of loci involved (Kearsey and Farquhar 1998). With this in mind, we produced Table 2, with L = 1250, μ = 10^{5}, m = 0.2, and V = 20. With these values, the mutational variance is equal to 10^{3}. Again, the first row gives the results for an unchanging environment. Note that, in this case, the level of heritability achieved is much higher
In their computer simulation study, Bürger and Lynch (1995) also chose parameter values that resulted in a mutational variance of 10^{3}. However, to do this, they used a moderate number of loci and a very high (and probably unrealistic) rate of mutation (μ = 2 × 10^{4}). Presumably, the high mutation rate was required because of computational limitations. For purposes of comparison, we produced Table 3, which presents data produced using parameter values very similar to those used by Bürger and Lynch (L = 50, μ = 2 × 10^{4}, m = 0.224, and V = 9).
While it is certainly possible that many phenotypic traits are controlled by a large number of loci, this suggests that mutations typically have pleiotropic effects, affecting more than one trait that is under selection. (Otherwise, an unrealistically large number of loci would be required to influence development of the phenotype.) Furthermore, experimental evidence suggests that pleiotropy is common (Caspari 1952; Wright 1977; Barton and Turelli 1989; Kondrashov and Turelli 1992; Caballero and Keightley 1994; Wagner 1996). In addition, the deleterious side effects caused by pleiotropy seem to be largest for mutations that have a large effect on the trait under study (Barton and Turelli 1989; Caballero and Keightley 1994). Thus, even if the number of loci controlling a trait is relatively large, it may be that mutations that do not have substantial deleterious side effects are rare. Furthermore, such mutations may be associated with relatively low values of m. With this in mind, we calculated Table 4, which uses the parameter set L = 10, μ = 10^{5}, m = 0.1, and V = 20. These sorts of values may be appropriate if mutations that do not have substantial deleterious side effects occur at a small number of loci and have relatively small phenotypic effects. We chose these particular values because they facilitate comparison with Table 1, which uses the same parameter values, with one exception (in Table 1, m = 0.2).
We do not intend to suggest that the data presented in Table 4 represents a complete study of the effects of pleiotropy. Nevertheless, the results are of interest and are likely to give some hint of the outcome of a complete study. The first row shows the data for an unchanging environment. We see that the level of heritability for a sexual population is low
The effect of environmental change: Let us now modify the model presented above to incorporate a shift in the phenotypic optimum, z_{opt}. Say that, at time t, the value of the optimum is given by z_{opt} = αt, where α > 0. In the presence of a steady change in the value of the optimal phenotype, the population tends toward a steadystate situation, where the mean phenotypic value changes at exactly the same rate at the optimum phenotypic value. When the steady state is achieved, the mean phenotypic value does not lie at z_{opt}, but lags behind z_{opt} by an amount denoted by ξ_{S} (i.e., ξ_{S} is the difference between the optimum and the mean phenotype). At the same time, the genetic variance (V_{G}_{,S}), the heritability
Inspection of Table 1 shows that, in a sexual population, altering the rate of environmental change (α) can have a very large impact on genetic variance (V_{G}_{,S}) and heritability
When the rate of environmental change is increased, the lag of the phenotypic mean behind the phenotypic optimum (ξ_{S}) also increases (see Table 1). As a consequence of the dependence of ξ_{S} and V_{G}_{,S} on α, increases in α produce increases in mean death rate (D_{S}). The data in Table 1 also suggest that, with 10 loci under selection, even very high values of α (up to α = 1) can be tolerated as long as the species is capable of doubling its birth rate (which is not a severe requirement).
Table 2 shows the effect of environmental change when the number of loci under selection is relatively large (L = 1250, with the other parameters set as for Table 1). In this case, the amount of genetic variance and heritability is substantial even in an unchanging environment. However, the effect of changing the environment is similar to what occurs when L = 10, in that every tenfold increase in α studied produces about a threefold increase in V_{G}_{,S}. Increases in V_{G}_{,S} of a similar magnitude are shown in Tables 3 and 4 for other choices of parameter values. Note that, because of computational constraints, it was not possible to use the entire range of α values found in Table 1 in all of the other tables. Thus, for example, we were unable to run the α = 0.0001 case for the parameter values used in Table 2.
Comparing Tables 1 and 2, we see that large numbers of loci generally increase the death rate in a sexual population as long as the rate of environmental change is not too high. This is because many loci lead to a relatively high level of genetic variance. However, for a very high rate of environmental change (α = 10), a large number of loci apparently confers an advantage, allowing a much lower death rate than what occurs with 10 loci.
Comparing Tables 1 and 4 leads to the conclusion that a relatively low value of m (the standard deviation of mutant effects) seems to have little impact on the death rate of a sexual population when the rate of environmental change is relatively small. However, for α ≥ 0.1, the lower value of m used for Table 4 leads to a substantially higher death rate than what is observed in Table 1. Presumably, this is because largeeffect mutations that push genotypes toward the optimum are more rare when m is low.
The effect of asexuality: So far, we have assumed that all L loci recombine freely. Let us now consider what happens in a population that is identical to the one just described, except that reproduction is asexual so that there is no recombination or segregation. The relevant data are shown in columns 3, 5, 7, and 9 in Tables 1, 2, 3, 4, and in Figures 1 and 2. After a sufficiently long period of time has elapsed, the mean death rate of asexuals takes on a value that does not change over time. We denote this value by D_{A}. The genetic variance, heritability, and the difference between the phenotypic mean and the optimum phenotype also settle down to steadystate values, denoted by V_{G}_{,A},
Examination of the tables shows that, in a changing environment, asexuals generally have much lower levels of genetic variance and heritability than a sexual population would have in the same circumstances. High heritability facilitates adaptation (Fisher 1930). Thus, it is not surprising to see that the mean phenotype for asexuals generally lags considerably further behind the optimum than does a sexual population in the same circumstances (ξ_{A} > ξ_{S}). In Figure 2 the distributions of genotypic effects for sexuals and asexuals are plotted as functions of α. The difference in widths (variance) of the distributions and the difference in lags are clearly seen.
The lower heritability and larger lags characteristic of asexuals do not mean that their mean death rate will necessarily be larger than in an equivalent sexual population. A large genetic variance may enhance heritability, but it also means that many individuals are far from the phenotypic optimum, and this tends to increase death rates. Thus, for example, in Table 2 (for which L = 1250) asexuals have a lower death rate than sexuals when α = 0.01. However, for larger values of α, the death rate is lower for sexuals. Indeed, for every set of parameter values studied, a sexual population has a much lower death rate than an asexual population when the rate of environmental change (α) is sufficiently large.
It is commonly held that asexuals have a twofold fertility advantage (Maynard Smith 1978; Michod and Levin 1988; Pecket al. 1997). If this is so, sexuals should still be able to obtain an overall fitness superiority as long as they live more than twice as long as asexuals (2D_{S} < D_{A}) and thus have more than twice as many offspring as asexuals do over the course of their lifetimes. Therefore, we now describe the conditions under which this inequality is satisfied.
Let α^{*} represent the rate of environmental change, α, for which D_{A} is twice the value that it takes when α = 0 (i.e., in an unchanging environment). The value of α^{*} depends on the various parameters of the model, but it is particularly sensitive to m, the standard deviation of mutant effects. If m is small, then mutations that move genotypic values substantially toward the optimum are rare, and this results in a small value of α^{*}. With a large value of m, even rapid rates of environmental change can be tolerated, and so α^{*} is relatively large. Thus, for example, with the parameters used for Table 1 (L = 10, μ = 10^{5}, m = 0.2, and V = 20) we have α^{*} = 0.0664. Table 4 uses the same parameters as Table 1, except that the value of m is halved to m = 0.1. This leads to a halving of the value of α^{*}, to α^{*} = 0.0332. Further investigations (not shown in the tables) show that, if we decrease m to onefifth of its value in Table 1 (m = 0.04), α^{*} falls by almost five times (to α^{*} = 0.0136).
It is interesting to note that, while α^{*} is quite sensitive to changes in the value of m, it is remarkably insensitive to changes in the allelic mutation rate, μ, as suggested by the analytic approximations found in the appendix (Equation A5). Thus, for example, if L = 10, m = 0.2, and V = 20, then when μ = 10^{6} we have α^{*} = 0.0555, which is only 16% lower than the value found in Table 1, for which μ was 10 times higher.
Were sex to allow perfect adaptation to environmental change, D_{S} would remain unaltered as α is increased. Furthermore, when α is sufficiently close to zero, D_{S} ≈ D_{A} for all plausible parametervalue choices that we have studied. Thus, in the case of perfectly adaptable sexuals, 2D_{S} < D_{A} will be satisfied when α is just a little larger than α^{*}. In reality, sexuals are not perfectly adaptable. Nevertheless, for the cases we have studied, we find that the value of α that is sufficiently large to induce 2D_{S} < D_{A} is typically not greatly in excess of α^{*}, except when the number of loci under selection is very large (as in Table 2). Thus, it seems very likely that only modest conditions must be met for the average lifetime of sexuals to be twice that of asexuals.
We denote the value of α for which 2D_{S} = D_{A} as α^{**}. In every case we have studied, 2D_{S} < D_{A} whenever α > α^{**}. We find that, for the parameters used in Table 1, α^{**} = 0.0779, which is to say that, to produce a twofold viability advantage for sexuals, the environment must change at a rate that is only 17% faster than the rate that would be required if sexuals were perfectly adaptable. When the value of m is halved, there is only a 15% difference between α^{*} and α^{**} (see Table 4). When we decrease the value of m to m = 0.04 (and leave the other parameters as in Table 1), we have α^{**} = 0.0152, which means that only a 12% difference separates α^{*} and α^{**}. This also means that, in this case, sexuals can gain a twofold advantage when the environment changes by <1.5% of a phenotypic standard deviation per generation.
DISCUSSION
Taken at face value, the results presented here suggest that sexual populations can maintain a remarkably high level of fitness, even when the environment is changing quite quickly. This conclusion is in sharp contrast with the results of most other studies (but see Kondrashov 1984). One example of the type of results generally found in the literature comes from the study by Bürger and Lynch, who suggested that populations were unlikely to be able to change at a rate >10% of a phenotypic standard deviation per generation (Bürger and Lynch 1995). We find that, with a plausible choice of parameter values, the environment can change at a much faster rate than suggested by Bürger and Lynch without inducing very high death rates in a sexual population. For example, for the parameters chosen for Table 1, α = 1 results in a death rate that is only twice what is found in an unchanging environment. When α = 1, Table 1 gives a value of V_{G}_{,S} of 3.24, and this means that a phenotypic standard deviation is equal to
Extreme caution should be exercised before accepting the conclusion that natural populations are much more able to deal with environmental change than previously realized. Our knowledge of the genetic systems underlying quantitative traits is very incomplete (Barton and Turelli 1989; Bulmer 1989). Before firm conclusions can be made, experimentalists must learn much more about pleiotropy, genetic constraints, population structure, and effective population size, and more must be done to incorporate these factors into models. We discuss some of these complications below.
We find that low rates of environmental change (e.g., α = 0.01) can induce a very large increase in genetic variation in comparison with the case of an unchanging environment (α = 0). This finding may help to explain the high levels of heritability and genetic variation commonly found in natural populations. High heritability has been a puzzle because the best current estimates of the relevant genetic parameter values suggest that low heritability is likely when environments do not change over time (Turelli 1984; Bulmer 1989). However, our finding about the impact of low rates of environmental change is another case in which our results appear, at least at first glance, to contradict previously published results, which have generally suggested that rapid changes in the environment are required to induce such large increases in genetic variation (Bürger and Lynch 1995; Kondrashov and Yampolsky 1996).
What accounts for the differences between our results and those of previous studies of the effects of environmental change? To address this question, it is useful to develop an intuitive understanding of the relationship between environmental change and genetic variation in our model. Recall that, when the optimum changes over time, the phenotypic mean value tends to lag behind the optimum value. As a result of this lag, a subclass of mutations is beneficial as they tend to bring phenotypes closer to the optimum. When these beneficial mutations occur, some of the resulting mutant alleles rise in frequency. However, once the optimum has moved further, new alleles become beneficial, and alleles that were once beneficial become deleterious and fall in frequency. This “turnover” of alleles is very different to what apparently occurs in large populations in an unchanging environment, where a single allele can become common and all other alleles remain rare forever (Turelli 1984). Our calculations show that the turnover in allele frequencies induced by environmental change can generate very substantial genetic variability.
The idea that environmental change can lead to an increase in genetic variation is in accord with much of the experimental literature (e.g., see pp. 200202 in Bell 1997). However, environmental change does not always induce an increase in genetic variation, presumably because of small experimental population sizes combined with strong selection (see below). Despite the plausibility of a relationship between environmental change and genetic variation, a number of key studies have assumed that the amount of genetic variation is independent of the rate of environmental change (Lynchet al. 1991; Charlesworth 1993; Lynch and Lande 1993). This assumption can be justified a number of ways. For example, Charlesworth (1993) considers a genetic system that follows the assumptions of the infinitesimal model (Bulmer 1980). This implies that gene frequencies do not undergo any substantial change as a result of environmental fluctuations, and as a consequence, genetic variance does not increase with the rate of environmental change. The infinitesimal model assumes an infinite number of loci, each of which has infinitesimal effects. We feel that these assumptions are much less realistic than the ones made in the model presented here.
A study that did make similar assumptions to ours was carried out by Bürger and Lynch (1995). In accord with our results, Bürger and Lynch found that a changing environment can increase genetic variance. However, the increases in variance reported by Bürger and Lynch were much smaller than those reported here. This is true even for the parameter values used in Table 3, which are very similar to those used by Bürger and Lynch.
The main reason for the differences between our results and those of Bürger and Lynch seems clear. As stated above, Bürger and Lynch relied on simulation methods, and thus they were unable to consider large population sizes (the largest population size used was 512). They did, however, observe that, as population size increases, there is also an increase in the degree to which environmental change enhances genetic variation. With this in mind, we strongly suspect that the main difference between our results and those of Bürger and Lynch is accounted for by the difference in population size.
To facilitate comparison with other studies, we used a Gaussian function for our distribution of mutant effects. There is, however, some evidence that a more leptokurtic function is more appropriate for this purpose (Mukaiet al. 1972; Mackayet al. 1992; Ohta 1992; Keightley 1994). For this reason, some researchers have used a “reflected gamma” distribution, with parameter 1/2 (Keightley and Hill 1987, 1988). We carried out some preliminary studies using this sort of function, and we find that the change makes very little difference to the outcome of evolution for a sexual population. For an asexual population, the change to a reflected gamma has a more substantial effect, allowing somewhat higher levels of heritability (
We assumed that the population is so large that stochastic effects can be ignored. This means that, effectively, the population has infinite size. Thus, there is always a supply of mutations that tend to move genotypes in the direction of the optimum. With this in mind, it is worth asking whether any natural populations are likely to be large enough to exhibit the sort of behavior reported here. Unfortunately, we have not yet been able to develop methods that allow us to address this question directly. However, it is possible to make a few relevant observations.
Consider, for example, the 7th row of Table 1, corresponding to α = 0.1. This row shows that, in a very large sexual population, the average population member has a phenotype that deviates from the optimal phenotype by 1.88 environmental standard deviations. This implies that the average allelic effect deviates from the optimum allelic effect by 1.88/(2L) = 1.88/20 = 0.0940 environmental standard deviations. Thus, the distance between the average effect and the optimum is less than onehalf of the standard deviation of mutant effects (m = 0.2). In addition, from the value of V_{G}_{,S} given in Table 1 (V_{G}_{,S} = 1.07), we can calculate that the standard deviation of allelic effects is equal to 0.231. Thus, a large percentage (∼34%) of alleles are actually in advance of the optimum. These alleles are suboptimal, but many of them will become closer to optimal in the next generation, after the value of the optimum phenotype has changed. These considerations suggest that an extremely large population size would not be necessary to produce effects that are roughly similar to those reported in the 7th row of Table 1 for a sexual population. We would guess that a population size of ∼10 times the inverse of the allelic mutation rate would be sufficient (i.e., a population with ∼10^{6} members).
The situation is very different under asexuality. Consider, once again, the 7th row of Table 1. Note that, for asexuals, the optimum is nearly eight environmental standard deviations away from the phenotypic optimum. Furthermore, the standard deviation in genotypic values
In the effectively infinite population studied here, a population, whether sexual or asexual, can always keep pace with a shifting optimum. If the optimum moves quickly, the population mean may fall well behind. However, once there is sufficient separation between the optimum and the population mean, mutations that move phenotypes substantially in the direction of the optimum will be strongly favored by selection. Once the strength of selection on these beneficial mutations is sufficiently strong, the population mean can change at the same rate as the optimum.
In a real and finite population, the necessary mutations may not always be available. Thus, it is possible for a changing optimum to outrun the population, in the sense that the difference between the optimum and the population mean may increase each generation, leading to a longterm decrease in fitness and finally to extinction of the population. In light of the results presented here, we feel certain that the rate of environmental change sufficient for the optimum to outrun the population will typically be much smaller for asexual populations than in the case of sexuality.
What accounts for the difference between the behavior of sexual and asexual populations that was observed in this study? There are a variety of roughly equivalent ways to answer this question, but perhaps the most straightforward explanation was put forward by James Crow (1992). Crow pointed out that, in the absence of new mutations, the mean of an asexual population cannot change beyond the genotypic value of the individual with the most extreme genotype in the current population. On the other hand, in the absence of mutation, a sexual population may be able to evolve far beyond the phenotype of any individual present in the population by combining rare alleles drawn from different individuals. The situation is similar when mutations are allowed to occur. Furthermore, mutations that push genotypes toward the optimum will tend to be lost in an asexual population unless they occur in one of the few individuals that has a genotype closest to the optimum (Fisher 1930; Manning and Thompson 1984; Peck 1994). This is not the case in a sexual population. Even if a beneficial mutation occurs in a very unfit individual, it can escape from the genetic context in which it arose through mating and recombination, and it can thus contribute to the longterm adaptation of the population.
A study by Charlesworth (mentioned above) shows that a sexual population can obtain a large advantage over asexuals in a changing environment, even if environmental change is not very fast (Charlesworth 1993). This finding is in qualitative agreement with our results. However, Charlesworth made use of the infinitesimal model, and as we have explained, this model is quite unrealistic despite its advantages. Furthermore, the substantial advantage of sex shown by Charlesworth vanishes for many plausible parameter choices. For example, the advantage is very small (or absent) if mutations affecting the trait are sufficiently rare. We also note that our results are related to studies of pure directional selection (Hill 1982; Bürger 1993). However, these studies are very different from ours, as they make the assumption that no optimum phenotype exists.
Our results suggest a number of ways that sexual populations might be maintained despite competition from asexual populations. First, this could happen if multiple traits are subject to changing optima. We expect that, if the number of these traits is sufficiently large, then a very low rate of change in the optimum for each trait will be all that is required to overcome the “twofold cost of sex” (Maynard Smith 1978; Charlesworth 1993; Pecket al. 1997). Second, sex might be maintained if only a single trait is subject to a changing optimum as long as the rate of environmental change is sufficiently swift. Furthermore, as we have seen, the rate of change required may be very modest as long as m is small. In fact, the relevant value of m in typical realworld situations may be very low. This is because many largeeffect mutations have substantial deleterious side effects, and thus they should not be considered when m is calculated (Barton and Turelli 1989; Caballero and Keightley 1994).
Finally, even with a large value of m and a single trait subject to a shifting optimum, sex might be maintained by occasional catastrophes, where the rate of environmental change temporarily rises to a level that would doom any population were it to go on unabated. Our results show that a long period of very slow environmental change can lead to much higher levels of heritability in a sexual population, as compared to an asexual population. If a period of very slow change is followed by a catastrophe, this heritability difference could easily lead to the extinction of asexuals. With their greater genetic variability, sexuals can adapt more quickly to the environmental change, and thus they may survive the catastrophe.
Our results do not represent a definitive analysis of the process whereby populations adapt to changing environments. Such an analysis must await more data on the genetics underlying quantitative traits, more sophisticated mathematical techniques, and much more powerful computers. Nevertheless, the work presented here should bring closer the day when we understand how organisms deal with the challenge of a changing world.
APPENDIX
Asexual population: Consider an effectively infinite population of diploid asexual organisms that evolve in continuous time and have alleles with continuous effects (Kimura 1965). During the life of an organism, the processes of viability selection and offspring production occur. Mutations in offspring are taken to occur at the time of birth and offspring are assumed to mature instantaneously to adulthood. Selection occurs on a single phenotypic trait that is controlled by 2L alleles located at L loci. The probability that one or more of the alleles in an offspring differ from those of its parent is ≃2Lμ (assumed ⪡1), where μ is the allelic mutation rate. It is highly unlikely that more than one allele in an offspring suffers a mutation, so the distribution of mutant effects is accurately taken to be that of a single allele.
The genotypic value of the trait, G, is continuous and runs from ∞ to ∞. The equation for the distribution of G at time t, namely Φ_{A}(G,t), is obtained from the limit of discrete time and discrete genotypic effects and an equation similar to that found by Kimura (1965) is obtained. In contrast to Kimura’s procedure, however, we explicitly regulate the total population density, so it remains constant in time. This is achieved by having a genotypeindependent birth rate that is equal to the mean death rate.
The focus of this work is the effect of a changing environment. To incorporate a constantly changing environment into the calculation, the death rate of individuals with genotypic value G at time t is taken to be D(G  αt), with α the constant rate of change of the optimal phenotype and D(G) the death rate in a static environment. Φ_{A}(G,t) is found to obey
The form of the death rate follows from the assumption, made in the main text, that individuals in a static environment have a death rate that depends quadratically on phenotypic value z. As a consequence, when the phenotypic death rate is averaged over the environmental effect (a Gaussian random variable that is independent of G and has mean zero and variance V_{e}), the death rate of individuals with genotypic value G at time t is D(G  αt), where D(G) = 1 + V_{e}/(2V) + G^{2}/(2V). In what follows, we scale variables so that, without loss of generality, V_{e} = 1.
After a transient period in a constantly changing environment, numerical evidence indicates that Φ_{A}(G,t) settles down to a steadystate tracking solution—one that continuously follows, without change of shape, the optimal genotypic value. Such a solution depends on G and t only in the combination G  αt and it is useful to write
When α = 0 we arrive at the equation for the equilibrium distribution in a static environment. Let V_{s} = VD(0). If 2LμV_{s}/m^{2} ⪡ 1, the distribution may be found in the “house of cards” approximation (Turelli 1984):
When the environment changes (α = 0) the foregoing mutation selection balance results may be changed significantly because the “environmental force” in Equation A3, αdΨ_{A}(G)/dG, need not be small compared with the mutation contribution,
Generally, the solution of Equation A3 has to satisfy
Let E_{A}(Var_{A}) denote the expectation operator (variance) for the asexual population. For tracking solutions, ξ_{A} denotes the lag of the mean genotypic value behind the value of the optimum and is defined by E_{A}(G) = αt  ξ_{A}. In terms of
Derivation of approximate results: The derivation given in this section is somewhat technical and may be omitted at a first reading. To begin, we introduce a parameter Θ defined by Θ^{2} = 2V[(1  2Lμ) D_{A} D(0)]. When α = 0, Θ^{2} is negative (Waxman and Peck 1998). A changing environment modifies the balance of evolutionary forces and allows the new possibility that Θ^{2} is positive. We estimate that for LμV/m^{2} ⪡ 1, Θ^{2} becomes positive at an α of order
Next we assume that the smallness of the trait mutation rate, 2Lμ, means that at almost all G where Ψ_{A}(G) is appreciable, the environmental term αdΨ_{A}(G)/dG in Equation A3 dominates the mutation contribution
So far the parameter Θ is unknown but an approximation for Θ can be derived by writing Equation A3 as the integral equation
After some manipulations, we find that Equation A7 can be written as
To make further progress, we make the crude approximation of keeping only the leading Θ dependence of ln(K(αV/m^{3}, Θ/m)). After some work, we find, ln(K(αV/m^{3}, Θ/m)) = Θ^{4}m^{2}/(8α^{2}V ^{2}) +...; thus Equation A8 yields Θ^{4}m^{2}/(8α^{2}V^{2}) ≈ ln(m^{2}/(4D_{A}LμV)). It is clear that among other things, various logarithmic corrections have been omitted to obtain this result, and in this spirit, we drop the factor D_{A} within the argument of the logarithm, thereby allowing Θ to be given in terms of known quantities:
Last, we note that the Gaussian distribution predicts V_{G,}_{A} × ξ_{A} = αV and this approximately applies when α is not too small (α > 0.001).
Sexual population: Consider an effectively infinite population of diploid sexual organisms with L unlinked loci that undergo random mating. The death rate for individuals with genotypic value G at time t is, as for asexuals in a constantly changing environment, D(G  αt).
We assume the population has achieved a steadystate tracking distribution, with G  αt having a timeindependent distribution. To proceed further, we neglect correlations of genes both across and between loci. We estimate both types of correlations to be comparable in magnitude and Bulmer’s analysis of betweenloci correlations (Bulmer 1989) indicates that correlation effects are small when V_{G}_{,S} ⪡ V. Results presented in the tables, which are consistent with the neglect of correlations, yield genetic variances satisfying V_{G}_{,S} ⪡ V. Results for which this inequality does not hold, e.g., those yielding V_{G,}_{S} ≥ V/2, are included for aid of comparison, but the reader should note that they may require significant corrections due to neglected correlations.
The neglect of correlations leads to all 2L alleles under selection having independent and identical distributions. In particular this means that for all values of the allelic location, i, the combination x_{i}  αt/(2L) has a timeindependent distribution. For the mean and variance of this quantity, we write M_{1} = E_{S}(x_{i}  αt/(2L)), V_{1} = Var_{S} (x_{i}  αt/(2L), where E_{S}(Var_{S}) denotes the expectation operator (variance) for the sexual population.
Each allele may be treated as that of a single haploid locus existing in an averaged genetic background composed of the remaining alleles. The background quantities are present in a nontrivial way and we give some details of the calculations. Consider, e.g., those individuals with an allele at location i with effect x_{i} at time t. Their death rate follows by averaging D(G  αt) over the genetic background and yields the death rate D_{S}(x_{i}  αt/(2L) + (2L  1)M_{1}), where
By virtue of being a tracking solution, Φ_{S,1}(x_{i},t) depends on x_{i} and t only in the combination x_{i}  αt/2L and we write
The lag of the mean genotypic value behind the fitness optimum, ξ_{S}, is defined by E_{S}(G) = αt  ξ_{S}. In terms of Ψ_{S}(x) we have
The distribution of genotypic effects, Φ_{S}(G,t) is approximately Gaussian, because G is the sum of 2L approximately independent Gaussian random variables with identical distributions.
Acknowledgments
The authors are indebted to Jonathan Yearsley for valuable advice and assistance. This research was supported by Biotechnology and Biological Sciences Research Council grant 85/G11043.
Footnotes

Communicating editor: A. G. Clark
 Received November 3, 1998.
 Accepted July 19, 1999.
 Copyright © 1999 by the Genetics Society of America