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 G=∑i=12Lxi .

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 per-allele 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 f(y−x∗)=(12πm2)exp(−(y−x∗)22m2), (1) where m is the standard deviation of mutant effects.

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 easy-to-measure 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, m² ⪡ 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 (hS2 ) is defined by hS2=VG,S∕(1+VG,S) . In this case, hS2=0.00801 .

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 2Lμm², and for Table 1, 2Lμm² = 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 (hS2=0.508) than what was seen in the first row of Table 1.

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 (hS2=0.00767) . However, this is largely a result of the low mutation rate, not the low value of m (Turelli 1984). Nevertheless, changing the value of m does have a substantial effect when the environment changes over time, as we show below.

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 steady-state 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 (hS2) , and the mean death rate (D_S) all depend on the rate of environmental change (α).

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 (hS2) . This finding is in accord with analytic approximations, which are presented in Equation A13 of the appendix. Over the range of values of α considered in Table 1 (0.0001 ≤ α ≤ 10), each tenfold increase in α produces about a threefold increase in V_G,S. As a consequence, the value of hS2 also increases dramatically with α. When α = 0.0001, we have hS2=0.0351 , which is low in comparison to values typically measured in natural populations (Lande 1975; Turelli 1984; Bulmer 1989). However, when α > 0.001, we have hS2>0.1 . Thus, as long as the optimum is changing by at least one phenotypic standard deviation every 1000 generations, it is possible to produce an appreciable level of heritability with only 10 loci. (We use the word “generation” to denote the average lifetime of a phenotypically optimal individual.)

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 large-effect 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 steady-state values, denoted by V_G,A, hA2 , and ξ_A, respectively. Analytic approximations for V_G,A and ξ_A are given in Equation A5 of the appendix.

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Figure 1.

—Effect of the rate of environmental change (α) on mean death rate. Circles give the data for sexuals (the D_S values), while squares give the data for asexuals (the D_A values). For both sexuals and asexuals, parameters were set as follows: L = 10, μ = 10^-5, m = 0.2, and V = 20.

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.

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Figure 2.

—Genotypic distributions as a function of the rate of environmental change (α). A genotype is characterized by its “genotypic value,” G, which is the mean phenotypic value for individuals with that genotype. The axis labeled G - αt gives the difference between genotypic values and the optimal genotypic value (αt). The higher ridge is the peak of the distribution for asexuals, for different values of α, while the lower ridge is the peak of the distribution for sexuals. Note that, on average, sexuals are closer to the optimal genotypic value than asexuals, and this difference increases with α. Also, both distributions become wider and lower as α increases. For both sexuals and asexuals, parameters were set as follows: L = 10, μ = 10^-5, m = 0.2, V = 20. The distributions shown are achieved after a sufficiently long period of time has elapsed.

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 one-fifth 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 parameter-value 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.

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 genotype-independent 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 ∂ΦA(G,t)∂t=[(1−2Lμ)D‒A(t)−D(G−αt)]ΦA(G,t)+2LμD‒A(t)∫−∞∞f(G−Y)ΦA(Y,t)dY. (A1) Here D‒A(t)=∫−∞∞D(G−αt)ΦA(G,t)dG is the mean death rate at time t and f(G) is given in Equation 1. The regulation of population density results in D_A(t) appearing as a factor of the final term in Equation A1.

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) + G²/(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 steady-state 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 ΦA(G,t)=ψA(G−αt). (A2) Substituting Equation A2 into Equation A1 and changing the variable to G′ = G - αt leads to an equation for Ψ_A(G′). For typographical simplicity we omit all primes and obtain −αdψA(G)dG=[(1−2Lμ)D‒A−D(G)]ψA(G)+2LμD‒A∫−∞∞f(G−Y)ψA(Y)dY, (A3) where D‒A=∫−∞∞D(G)ψA(G)dG .

When α = 0 we arrive at the equation for the equilibrium distribution in a static environment. Let V_s = VD(0). If 2LμV_s/m² ⪡ 1, the distribution may be found in the “house of cards” approximation (Turelli 1984): ψA(G)≡ΦA(G)≈4LμVsf(G)∕(G2+β2),β=4πLμVs∕2πm2 . This distribution has a mean of zero and an approximate variance of 4LμV_s. Alternatively, if 2LμV_s/m² ⪢ 1, Φ_A(G) is approximately Gaussian, with mean zero and variance 2LμVsm2 . This result is derived using a similar approach to that of Kimura (1965).

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, 2LμD‒A∫−∞∞f(G−Y)ψA(Y)dY .

Generally, the solution of Equation A3 has to satisfy ψ(G)≥0,∫−∞∞ψ(G)dG=1 (A4) as follows from Φ_A(G,t) being a probability density. Numerical evidence supports the assumption that the solution of Equation A3 with these properties is unique.

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 ψA(G),ξA=−∫−∞∞GψA(G)dG and the variance of genotypic effects is VG,A=∫−∞∞[G−(−ξA)]2ψA(G)dG . The mean death rate following from these is D‒A=1+1∕(2V)+(VG,A+ξA2)∕(2V) (recall that we use units where V_e = 1). Because there is no epistasis or dominance in the model, the narrow sense heritability is the ratio of genotypic to phenotypic variance: hA2=VG,A∕(VG,A+1) . Tables 1, 2, 3, 4 contain results following from the numerical solution of Equation A3. The numerical results are complemented by an analytical approximation that applies for asexuals when LμV/m² ⪡ 1: ξA≃αV∕m[8ln(m24LμV)]1∕4,VG,A≃αmV[8ln(m24LμV)]−1∕4. (A5) These approximations exhibit the general dependencies of the lag, ξ_A, and the genetic variance, V_G,A, on parameters in the problem. In particular, they indicate that as far as the rate of optimum shift, α, is concerned, both the lag and the genetic variance are proportional to α . Furthermore, the approximations show that ξ_A and V_G,A exhibit a strikingly weak dependence on the mutation rate, μ.

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 Θ² = 2V[(1 - 2Lμ) D_A -D(0)]. When α = 0, Θ² is negative (Waxman and Peck 1998). A changing environment modifies the balance of evolutionary forces and allows the new possibility that Θ² is positive. We estimate that for LμV/m² ⪡ 1, Θ² becomes positive at an α of order α0≡def27(2Lμ)3V2m−3 . Here we assume that α is sufficiently large that Θ² > 0. Furthermore, we take Θ itself to be positive.

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 2LμD‒A∫−∞∞f(G−Y)ψA(Y)dY . We thus neglect the latter term where Ψ_A(G) is appreciable, in which case −αVdψA(G)∕dG≈12(ϴ2−G2)ψA(G) . Only in the vicinity of G = -Θ is Ψ_A(G) appreciable, because it has a maximum at this point. Approximating ½(Θ² - G²) by Θ (Θ + G) yields a Gaussian approximation for ψA(G):ψA(G)≈ϴ∕(2παV)exp[−(ϴ∕(2αV))(G+ϴ)2] . This is a distribution with a lag of ξ_A = Θ and a variance of V_G,A = αV/Θ.

So far the parameter Θ is unknown but an approximation for Θ can be derived by writing Equation A3 as the integral equation ψA(G)−2LμD‒A∫−∞∞R(G,Y)ψA(Y)dY=0, (A6) where R(G,Y)=(1∕α)∫G∞exp(−[H(G)−H(X)]∕α) . An approximate equation can be obtained by expanding the Fredholm determinant associated with Equation A6. The simplest nontrivial approximation keeps only terms up to linear order in R and yields (Lovitt 1924) 1−2LμD‒A∫−∞∞R(G,G)dG≈0. (A7) For α → 0, it may be shown that the value of Θ² obtained from this equation coincides with the house of cards approximation (Turelli 1984), thereby indicating that the approximation holds when LμV/m² ⪡ 1.

After some manipulations, we find that Equation A7 can be written as 1−4D‒ALμVm2K(αVm3,ϴm)≈0,K(λ,q)=∫0∞exp[−λ2(x624+x42)+q2x22]dx. (A8)

To make further progress, we make the crude approximation of keeping only the leading Θ dependence of ln(K(αV/m³, Θ/m)). After some work, we find, ln(K(αV/m³, Θ/m)) = Θ⁴m²/(8α²V ²) +...; thus Equation A8 yields Θ⁴m²/(8α²V²) ≈ ln(m²/(4D_ALμV)). 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: ϴ≈αV∕m[8ln(m2∕(4LμV))]1∕4 . Analytical approximations for ξ_A and V_G,A then follow from this formula for Θ.

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 steady-state tracking distribution, with G - αt having a time-independent 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 between-loci 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 time-independent distribution. For the mean and variance of this quantity, we write M₁ = E_S(x_i - αt/(2L)), V₁ = 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)M₁), where DS(x)≡def(1+12V)+(2L−1)V12V+x22V. (A9) The distribution of allelic effects at location i, written as Φ_S,1(x_i,t), satisfies the one-locus haploid equation ∂ΦS,1(xi,t)∂t=[(1−μ)D‒S−DS(xi−αt2L+(2L−1)M1)]⋅ΦS,1(xi,t)+μD‒S∫−∞∞f(xi−y)ΦS,1(y,t)dy, where D‒S=∫−∞∞DS(xi−αt2L+(2L−1)M1)ΦS,1(xi,t)dxi, a quantity independent of time.

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 ΦS,1(xi,t)=ψS(xi−αt2L+(2L−1)M1). (A11) Changing the variable to x = x_i - αt/(2L) + (2L - 1)M₁ in Equation A10 yields −α2LdψS(x)dx=[(1−μ)D‒S−DS(x)]ψS(x)+μD‒S∫−∞∞f(x−y)ψS(y)dy. (A12)

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 ξS=−ES(G−αt)=−∫−∞∞[xi−αt∕(2L)+(2L−1)M1]ΦS,1(xi,t)dxi=−∫−∞∞xψS(x)dx . Similarly, the total genotypic value for the sexuals is VG,S=VarS(Σi=12Lxi)=2LV1andV1=∫−∞∞[x−(−ξS)]2ψS(x)dx . The mean death rate and narrow sense heritability are D‒S=1+1∕(2V)+(VG,S+ξS2)∕(2V),hS2=VG,S∕(VG,S+1) . Tables 1, 2, 3, 4 contain results following from the numerical solution of Equation A12. These are complemented by analytical approximations that follow from the same methods as those used for the asexuals. The approximations apply when μV/m² ⪡ 1: a fairly unrestrictive condition. The approximations for sexuals can be obtained from the asexual results of → Equation A5 by the replacements ξ_S = ξ_A(α → α/(2L), μ → μ/(2L, V_G,S = 2L · V_G,A(α → α/(2L), μ → μ/(2L), as follows from comparing and A3: ξS≃αV∕(2Lm)[8ln(m22μV)]1∕4,VG,S≃2LαmV[8ln(m22μV)]−1∕4. (A13) As for asexuals, the approximations indicate that both the lag and the genetic variance are proportional to α and there is a very weak dependence on the mutation rate, μ. Also, as for asexuals, the approximations yield the relation V_G,S × ξ_S = αV, which approximately applies when α is not too small (α > 0.001).

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.