The distribution of the final gap δ_f:

The final gap δ_f is the phenotypic gap right before an adaptive step. Obviously, it is determined by the initial gap δ_i and by the time at which the step occurs. However, it is convenient to frame our analysis directly in terms of δ_f, with the time between steps entering only implicitly. In appendix a, we derive the distribution of δ_f conditioned on δ_i. In particular, we obtain the complementary distribution function F_γ(δ_f | δ_i), that is, the probability that the final gap is greater than δ_f for given δ_i. We find F_γ(δ_f | δ_i) = 1 for δ_f < δ_i and(8a)for δ_f ≥ δ_i, where(8b)and(8c)

Explicit expressions for g(δ) with particular choices of p(α) are given in appendix a. Note that the dependence of F on the ecological and genetic model parameters can be summarized in the single composite parameter γ. As we discuss below, this parameter plays a key role for the characteristics of the adaptive process. Differentiation of (8) yields the corresponding density function, which is(9)for δ_f ≥ δ_i and f_γ(δ_f | δ_i) = 0 for δ_f < δ_i.

The conditional distribution of the step size α:

Assume now that a step occurs when the total gap equals δ_f. The probability that it has size α is proportional to the fixation probability of the respective mutation, 2p(α)s(α | δ_f), where(10)is the selection coefficient given δ_f (see Equations 6 and 7). As only mutations with a positive selection coefficient (α between 0 and 2δ_f) can result in a substitution, we obtain the density of α given δ_f as(11)

An adaptive walk following Equation 5 can be easily simulated by specifying the initial gap for the first step and then iteratively applying (i.e., drawing from) Equations 8 and 11. For further progress, however, we need to say something about the distribution of the initial gap values δ_i over the course of the adaptive walk.

The distribution of the initial gap δ_i:

The sequence of initial gaps δ_i can be viewed as a Markov chain. From (9) and (11), and using (5), the distribution of the initial gap, ρ(δ_i), evolves according to(12a)where the transition density R is given by(12b)The integration runs over all final gap values δ_f that are larger than the old initial gap δ_i and in absolute value larger than the new initial gap . Since the transition density R depends only on γ [and on the distribution of new mutations, p(α)], this must also hold for any n-step distribution of δ_i and, in particular, for the equilibrium distribution (δ_i). In extension, the same is true for the distributions of δ_f and α. In the limit γ → 0, we obtain R_γ→0( | δ_i) ∝ p(δ_i – )( – ()²) for || ≤ |δ_i| and R_γ→0( | δ_i) = 0 otherwise. The equilibrium distribution for the initial gap ρ_γ→0^(∞)(δ_i) then converges to a point measure with weight 1 at δ_i = 0 (i.e., the gap becomes infinitely small).

The unconditional distribution of the step size α:

We can express the unconditional distribution of step sizes α as(13a)where(13b)is the conditional distribution of α given the initial gap δ_i. To derive the unconditional distribution ϕ_γ(α), we need to decide on a distribution ρ_γ for the initial gap. In general, it is not possible to solve Equation 12a for the equilibrium distribution . For an analytical approximation of ϕ_γ(α), we instead focus on the distribution of δ_i at the beginning of the adaptive process. In particular, we assume that the population is initially (at the time the optimum starts moving) perfectly adapted. Then the distribution of the initial gap before the first step, (δ_i), is a point measure with weight 1 at δ_i = 0, and (13) evaluates to(14)for α ≥ 0 and (α) = 0 for α < 0. For a refined approximation, we can apply Equation 12 to (δ_i) and derive the distribution of the initial gap before the second step, (δ_i), and from that, the distribution of the size of the second step, (α). As we show below, and are good approximations for the equilibrium distribution .

The distribution of fitness effects at the time of fixation:

In addition to the phenotypic effects of adaptive substitutions, we might also be interested in their distribution of fitness effects (see Gillespie 1983; Gerrish and Lenski 1998; Orr 2002, 2006; Rozen et al. 2002; Martin and Lenormand 2008), where fitness is measured at the time the substitution takes place. To calculate this distribution, we first need to derive the distribution of fitness effects of new mutations for a given final gap δ_f. Due to the symmetry of the fitness function (10), there are generally two α-values for any selection coefficient s (corresponding to under- and overshooting of the optimum),(15)provided that . The cumulative distribution function (cdf) of the fitness effects of new mutations given δ_f is(16)where A and S are random variables for the step size and the selection coefficient, respectively, and P(α) is the cdf of the phenotypic effects of new mutations [the integral of p(α)]. Differentiation yields the corresponding density(17)for s ≤ σ and 0 otherwise. Here, the denominator is the magnitude of the partial derivates of α_1,2 with respect to s. In analogy to Equation 13b, and focusing on the first step of the adaptive walk (i.e., δ_i = 0), the distribution of fitness effects of adaptive substitutions is then(18)for 0 < s ≤ σ and 0 otherwise. Further results for the distribution of fitness effects are given in appendix c.

Analytical results:

Figure 2 shows the distribution of adaptive substitutions in the adaptive-walk approximation for two different values of the parameter γ (Equation 8c) and for different stages of the adaptive process. The shaded areas show the equilibrium distribution , obtained from numerical iteration of Equations 8 and 11. is bimodal with a large positive and a very small negative part (indicating that backward steps are rare). In addition, the solid line in Figure 2B shows the distribution of the first step, , which is very close to for α ≥ 0, but does not extend to α < 0. Backward steps are, however, included in the distribution of the second step, , which is virtually indistinguishable from the equilibrium distribution (Figure 2C). In the following, we often neglect backward steps and use as a useful first approximation to .

Comparing the left and right columns of Figure 2, we see that ϕ_γ depends strongly on the composite parameter γ = v/(σΘ). To understand the role of this key parameter, it is instructive to rewrite Equation 8c as(19)Here, v and σ are our usual model parameters, which are measured in units of the mutational standard deviation ω (see model description), and and are the unscaled counterparts. The unscaled version in (19) makes explicit the strong dependence of γ on the width of the distribution of new mutations (γ ∼ 1/ω³). Since γ itself is dimensionless, it does not depend on a particular scale. It relates two key determinants of the model behavior: While the numerator captures the dynamics of the environment, the denominator can be understood as a measure for the “adaptive potential” of the population, that is, its ability to follow the moving optimum. The adaptive potential includes both genetic factors (the mutation rate and the distribution of new mutations) and the static component of the selective environment (the selection strength σ). If γ is small, the environment changes slowly relative to the adaptive potential. This suggests that the adaptive process will be determined primarily by the environmental dynamics. In contrast, for large γ, the adaptive process is determined mainly by genetic factors, especially by the distribution of new mutations (see below). To make these ideas more explicit, we now formalize the concept of the selective sieve (see Introduction).

The selective sieve:

According to Equation 13, the distribution of adaptive substitutions is the product of three factors:(20a)Here, p(α) is the distribution of new mutations, which provides the raw material for adaptation. Ψ(α, γ) constitutes the selective sieve, which expresses the average fixation probability over the adaptive process and transforms new mutations into substitutions. As a function of α, the selective sieve splits into two basic components (compare Equation 13):(20b)(20c)(where ∝ indicates proportionality; all factors not depending on α are ignored).

The first component Ψ_s is the static sieve, which is independent of the ecological and genetic dynamics. It reflects the fact that (in our additive model) mutations with small phenotypic effects can have only small fitness effects and, hence, are often lost due to genetic drift. In consequence, the static sieve favors large adaptive steps, which is the classical argument raised by Kimura (1983) in the context of Fisher's geometric model.

The second sieve component Ψ_d is the dynamic sieve. Unlike the static sieve, it depends on γ and accounts for the dynamic changes in both the population and the environment. It is the dynamic sieve that removes mutations overshooting the optimum by more than the final phenotypic gap δ_f (see Figure 1B). Thus, in contrast to the static sieve, the dynamic sieve favors small mutations over large ones.

The three factors in Equation 20 combine to determine the shape of the distribution of adaptive substitutions (Figure 3). Clearly, the substitution rate of small mutations is limited by the static sieve Ψ_s. As new mutations with small phenotypic effects have low fixation probabilities, ϕ_γ(α) decreases linearly to zero for |α| → 0. This holds even if small mutations are abundant (as is the case for the exponential distribution of new mutations), as long as their density is bounded at α = 0 (since also the dynamic sieve Ψ_d is always bounded at α = 0).

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Figure 3.—

Illustration of the selective sieve (Equation 20). The plots show the factors contributing to the distribution of adaptive substitutions [or, more precisely, to the distribution of the first step, (α), Equation 14] for two values of γ = v/(σΘ). To derive the distribution of adaptive substitutions, the distribution of new mutations p(α) (thin solid line) is first multiplied by the static sieve Ψ_s (dotted line, Equation 20b), yielding the limiting distribution ϕ_∞(α) (shaded line, Equation 24). Multiplication of ϕ_∞(α) by the dynamic sieve Ψ_d (dashed line, Equation 20c) gives the distribution of adaptive substitutions ϕ_γ(α) (thick solid line). The static and dynamic sieve components are defined only up to proportionality. Here, Ψ_s is scaled such as to be an asymptote to ϕ_∞, and Ψ_d such as to yield 1 at the intersection of ϕ_∞ and ϕ_γ. Note that the static sieve is independent of γ, whereas the dynamic sieve is less effective (i.e., less steep) if γ is large.

In contrast, the substitution rate of large mutations is limited by both the dynamic sieve Ψ_d and the distribution of new mutations p(α). As shown in Figure 3, the relative importance of these two factors depends on γ. For small γ (Figure 3A), the right-hand tail of ϕ_γ(α) is determined mainly by the dynamic sieve, which sharply cuts off the right-hand tail of p(α). In contrast, for large γ (Figure 3B), the dynamic sieve is weak (i.e., Ψ_d decreases only slowly with α), and the right-hand tail of ϕ_γ(α) is determined primarily by p(α). For intermediate γ, both factors are important. In appendix b, we show that they interact in a complex way, whose details depend on the fourth moment of p(α).

Genetic vs. environmental limitation:

The way in which γ determines the relative importance of the dynamic sieve and the distribution of new mutations in shaping the distribution of adaptive substitutions is a key result of this article. Indeed, the two examples shown in Figure 3 can be generalized to describe two different modes of adaptive evolution, or adaptive regimes, in the moving optimum model. In accordance with our interpretation of the parameter γ (see above, Equation 19), we say that the adaptive process (which can be characterized by the distribution of adaptive substitutions) is environmentally limited if γ is small and the right-hand tail of ϕ_γ(α) is mainly determined by the dynamic sieve. In contrast, we say that the adaptive process is mutation limited if γ is large and the right-hand tail of ϕ_γ(α) is mainly determined by the distribution of new mutations. These two cases are best viewed as the ends of a continuum. However, they can be studied in a “pure” form by considering the limits of very small or very large γ, in which only one of the two factors, Ψ_d or p(α), is operating.

Large γ—mutation-limited regime:

In the opposite extreme of large γ, the environment changes fast relative to the adaptive potential of the population. As a consequence, the phenotypic gap δ_f will typically be large, and large mutational steps can contribute to the adaptive process. In this case, the dynamic sieve is not effective, and ϕ_γ(α) is limited for large α by the distribution of new mutations p(α). In the limit γ → ∞, Ψ_d(α) is constant. Only the static component of the sieve, Ψ_s(α), remains in place, and the distribution of adaptive substitutions satisfies(23)

Normalization yields the solution(24)where μ is the average size of a positive new mutation (α > 0) under the distribution p(α). Since Equation 24 is independent of the initial gap δ_i, it holds for all steps in the adaptive walk. For an exponential distribution of new mutations, (24) is a Gamma distribution with shape parameter 2 (Rozen et al. 2002). Furthermore, since , the mean step size evaluates to(25)

If the optimum increases indefinitely (as is assumed here), the limit γ → ∞ (e.g., v → ∞) might appear unrealistic. However, Equation 24 also holds if the optimum stops moving once the phenotypic gap has reached a value δ_f sufficiently larger than μ. In this case, γ → ∞ corresponds to a sudden change in the environment (see Simulation results below). Indeed, an alternative derivation of ϕ_∞(α) is as a limit of the conditional distribution ϕ(α | δ_f) (Equation 11) for large δ_f (in which case s ∝ α). It is worth noting that Equation 24 corresponds to the distribution of the first substitution predicted by Kimura (1983) for Fisher's geometric model (see also Orr 1998).

For an exponential distribution of new mutations, the distribution of adaptive substitutions is well approximated by the limiting distribution if γ 0.1 and by the limiting distribution ϕ_∞ if γ 10 (Figure 4). For γ ≈ 1, the distribution of adaptive substitutions is between those extremes, and the mean step size is less than that predicted by either of the two limits (Figure 4F). A more in-depth analysis of the transition between the environmentally and mutation-limited regimes is given in appendix b.

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Figure 4.—

Dependence of the distribution of adaptive substitutions, ϕ_γ(α), on the composite parameter γ = v/(σΘ) (solid line). (A–E) The shaded area is the equilibrium distribution of the adaptive-walk approximation, as obtained from 100,000 iterations of Equations 8 and 11. The solid line is the distribution of the first step, (α), according to Equation 14. The other two lines show the limiting distributions discussed in the main text: The dotted line is the distribution (Equation 21), which is a good approximation in the environmentally limited regime (small γ). The dashed line is the distribution ϕ_∞ (Equation 24), which is the limiting distribution in the mutation-limited regime (large γ). (F) The mean step size for all three distributions as a function of γ (where the dotted line is from Equation 22 and the dashed line is from Equation 25).

The distributions of the first steps and in equilibrium:

Finally, we can explain why the equilibrium distribution of adaptive substitutions is usually well approximated by the distributions of the first and second steps (see Figure 2). The explanation depends on whether the adaptive process is environmentally or mutation limited. This is because and have the same limit distributions for either γ → ∞ or , respectively (see above).

If γ is small and evolution is environmentally limited, the distribution of adaptive substitutions is strongly influenced by the dynamic sieve Ψ_d. In particular, the conditional distribution ϕ_γ(α | δ_i) (Equation 13b) depends strongly on the initial phenotypic gap δ_i (Figure 5A1). However, in this case, the gap typically remains small (Figure 5B1) and can be approximated by δ_i = 0.

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Figure 5.—

The role of the initial phenotypic gap δ_i in the adaptive-walk approximation. (A) The shaded area shows the equilibrium distribution of adaptive substitutions, (α) (same data as in Figure 2). The lines are the conditional distributions ϕ_γ(α | δ_i) given the initial phenotypic gap δ_i (Equation 13b). The shaded line (negative α) is for δ_i = −0.5 in A1 and for δ_i = −1 in A2. (B) The equilibrium distributions of the initial and the final gap, along with the distribution of new mutations (reflected exponential with ω = 1, Equation 4).

In the opposite case of large γ (mutation-limited regime), the initial (and final) phenotypic gap can be substantial (Figure 5B2). Equation 13 shows that the only effect of the distribution of the initial gap, ρ(δ_i), on ϕ_γ(α) is through the dynamic sieve Ψ_d. As discussed above, however, the dynamic sieve is not effective in the mutation-limited case, and ϕ(α) is limited by the distribution of new mutations. Similarly, the value of δ_i has little influence on the conditional distribution ϕ_γ(α | δ_i) (Figure 5A2).

In summary, for small γ the phenotypic gap is small and can be approximated by δ_i = 0, whereas for large γ, the gap is large but has little effect on the distribution of adaptive substitutions. In consequence, the distribution of step sizes during an adaptive walk is not confounded by transient nonequilibrium effects. As long as the decisive parameter γ is kept constant, the equilibrium distribution is reached almost immediately and is well predicted by the distribution of the first (or second) step.

Simulation results:

In the following, we compare the analytical predictions from the adaptive-walk approximation to results from individual-based simulations. In contrast to the previous section, we use the unscaled parameters and introduced in Equation 19. Also the substitution effects are measured on a given constant scale, that is, not relative to ω. Using this constant scale makes explicit that an increase in γ due to a decrease in ω (see Equation 19) leads to a decrease in the average absolute substitution effect , whereas the relative effect α always increases with γ (compare Figures 4 and 6).

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Figure 6.—

The distribution of adaptive substitutions in diploid and haploid populations experiencing a low influx of mutations (Θ = 0.2). Histograms show the observed distribution obtained from individual-based simulations. The solid line is the prediction for the first step from the adaptive-walk approximation (Equation 14). The dashed line is the distribution of new mutations p(α) (reflected exponential with standard deviation ω, Equation 4). The value in parentheses is the composite parameter γ (Equation 19). Other parameters: , , L = 10, K = 1000, u = 10⁻⁵, Θ = 2LKu. Note that, because mutational effects are measured on an absolute scale (not relative to ω), the mean step size decreases with γ (whereas it increases on the relative scale, see Figure 4).

Figure 6 shows that the predictions from the adaptive-walk approximation are very accurate for populations experiencing a low influx of mutations (small Θ). A comparison with Figure 2 shows that backward steps are even rarer than in the adaptive-walk approximation. Therefore, the equilibrium distribution of adaptive substitutions is well predicted by the distribution of the first step, ϕ_γ⁽¹⁾ (Equation 14). This is true for various shapes of the distribution of new mutations (including Gaussian, Gamma, and uniform distributions, data not shown) and in both the environmentally limited (large ω, small γ) and mutation-limited (small ω, large γ) regimes. Furthermore, the result holds in both haploid and diploid populations.

In contrast, if the influx of mutations into the population is high (large Θ), the observed distribution of adaptive substitutions deviates from the adaptive-walk prediction (Figure 7), and the direction of the deviation depends on the genetic architecture. For given Θ, substitutions are larger than predicted if the number of loci, L, is small but smaller than predicted if the number of loci is large (assuming the total individual mutation rate Lu is kept constant).

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Figure 7.—

Effect of the population mutation rate Θ and the number of loci L on the distribution of adaptive substitutions. Histograms show the observed distribution obtained from individual-based simulations of a diploid population, assuming the distribution of new mutations is reflected exponential (Equation 4) with . The solid line is the prediction for the first step from the adaptive-walk approximation (Equation 14). Deviations between observation and prediction are due to linkage for small L and due to epistasis for large L (see text). In each row, Θ was varied by increasing K (10³, 10⁴, 10⁵), while in each column, u was adjusted to keep the product Lu constant. Parameters: Θ = 2LKu, , . The composite parameter γ (Equation 19) = 2.21 in A, 0.221 in B, and 0.0221 in C.

To explain these findings, recall that, in the adaptive-walk approximation, fixation (or loss) of mutant alleles is instantaneous and, therefore, adaptation occurs in well separated steps. In populations with large Θ, this assumption is violated. Instead, at any given time, different mutant alleles are likely to be segregating simultaneously. These alleles interact with each other and compete for fixation. The important point is that they do so in two different ways. One is based on linkage and the other one on epistasis.

Linkage effects:

If two mutations get established in different genetic backgrounds, the only way for them to go to fixation together is by a recombination event that brings them onto the same background. Otherwise, one of them will eventually outcompete the other. This effect has been described as the Hill–Robertson effect for sexual populations (Hill and Robertson 1966; Barton 1995) and as clonal interference for asexual populations (e.g., Gerrish and Lenski 1998; Park and Krug 2007). In Figure 7, the linkage effect is strongest for small L, because decreasing L (while keeping Θ constant) amounts to reducing recombination. The effect of recombination is also investigated in Figure 8, where the recombination rate r between adjacent loci is varied.

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Figure 8.—

Effect of linkage on the distribution of adaptive substitutions in populations experiencing a high influx of mutations (Θ = 20). r is the recombination rate between adjacent loci. Histograms show the observed distribution obtained from individual-based simulations of a haploid population, assuming the distribution of new mutations is reflected exponential (Equation 4) with . The solid line is the prediction for the first step from the adaptive-walk approximation (Equation 14). Parameters: K = 10⁵, L = 10, u = 10⁻⁵, Θ = 2LKu, , . The composite parameter γ = 2.2 × 10⁻⁵ (Equation 19).

Both Figures 7 and 8 show that, in the moving optimum model, increased linkage (interference) favors mutations with large phenotypic effect. Two factors contribute to this phenomenon. First, interference leads to a reduced substitution rate (because some mutations are lost even after they have survived drift), which increases the average gap between the mean phenotype and the optimum and leads to mutations with larger average effect being recruited. Second, a mutation with large phenotypic effect is more likely to outcompete an interfering small one than vice versa, even if both mutations initially have the same selective advantage. The reason is that, with a moving optimum, the selection coefficient of the large mutation will increase faster (see Equation 7).

As shown in Figure 9, the importance of the linkage effect increases with the rate of environmental change ṽ, but is relatively insensitive to the selection strength . This can be explained as follows: The amount of interference depends on the probability of cosegregating mutations, which is determined by the substitution rate (the number of substitutions per time) and the average fixation time T_f (i.e., the time a mutation takes to increase from a single copy to frequency 1, cf. Kopp and Hermisson 2009). Increasing ṽ leads to a strong increase in the substitution rate, because adaptation needs to cover a larger phenotypic distance per time, whereas the mean step size increases only relatively weakly (see Figure 4F). It also leads to a decrease in the average fixation time (because overall selection is stronger), but this is not enough to compensate for the increased substitution rate. In consequence, more alleles are segregating at any given time, and the opportunity for interference increases. In contrast, if ṽ is constant and increases, the substitution rate increases only slightly (because the average step size decreases), and this effect approximately cancels with the decreased fixation time due to stronger selection. Because the linkage effect is (more or less) independent of , it also has no strong relation to γ and can be equally strong in the environmentally and mutation-limited regimes.

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Figure 9.—

Effect of the (unscaled) selection strength and speed of environmental change on the distribution of adaptive substitutions in populations experiencing a high influx of mutations (Θ = 20). Histograms show the observed distribution obtained from individual-based simulations of a haploid population, assuming the distribution of new mutations is reflected exponential (Equation 4) with . The solid line is the prediction for the first step from the adaptive-walk approximation (Equation 14). Also shown is the value of the composite parameter γ (Equation 19) and the average fixation time of mutant alleles (see text). The main purpose of the figure is to illustrate how the deviation between observation and prediction, which stems from the linkage effect, depends on and . Parameters: K = 10⁵, L = 10, u = 10⁻⁵, Θ = 2KLu.

Limited increase of the optimum:

So far, we have assumed that the optimum increases indefinitely, which is, of course, an idealization of biological reality. To test the robustness of our results for shorter bouts of adaptation, we now assume that the optimum stops after reaching a maximal value z_max (see Simulation methods). To avoid complications caused by interference between cosegregating alleles (see above), we focus on populations with small Θ.

In the model with an indefinitely moving optimum, we have shown that the distribution of step sizes changes little over the course of the adaptive process. Indeed, a key result is that the long-term equilibrium distribution is very similar to the distribution of the first steps (Figure 2). In contrast, with a limited increase of the optimum, the size of the later steps is bound to decrease as the population approaches the final optimum. This is, indeed, observed in our simulations (right-hand columns in Figure 10).

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Figure 10.—

The distribution of adaptive substitutions if the optimum stops at z_max. The rows show results for three sets of simulations with different z_max but equal speed of the optimum . In the left-hand column, bar charts show the average distribution of step sizes over 100 replicated simulations until adaptation to the new optimum. The solid line is the prediction from the adaptive-walk approximation (Equation 14), which assumes an unlimited increase of the optimum. The dashed line shows the distribution of new mutations p(α) (reflected exponential with ; Equation 4). In the right-hand column, the solid line shows the average size of the first, second, etc., steps (left y-axis). Error bars are standard errors. The dashed line shows the mean phenotype of the population at the time of the respective step, measured relative to the final optimum z_max (right y-axis). The shaded area indicates the proportion of simulations that reached a given number of steps (right y-axis). Parameters: haploid population with L = 10, K = 10³, u = 10⁻⁵, Θ = 2LKu = 0.2, . The composite parameter γ = 0.0177 (Equation 19).

We can now ask for how long the optimum must move so that the distribution of adaptive steps is accurately predicted by the adaptive-walk approximation (which assumes an indefinite movement). As illustrated in Figure 10, this is the case if the ratio of the total movement of the optimum to the predicted average size of a single step, , is sufficiently large. Most steps will then occur during the moving phase, and the final steps during the constant phase have no significant effect on the overall distribution (Figure 10C). Our simulations indicate that a value of is sufficient for a good fit with the prediction from Equation 14. This result is independent of whether the ratio is large because z_max is large or because is small (additional results not shown).

The sudden change scenario assumed in most previous studies of adaptation can be seen as a limiting case of the moving optimum with v → ∞. In Analytical results, we have argued that the limiting distribution ϕ_∞(α) (Equation 24) is an approximation for the sudden change scenario, provided the jump of the optimum (which is equivalent to the final phenotypic gap δ_f) is large enough. This is illustrated in Figure 11. For a small jump, there is an excess of small and negative steps relative to the prediction from Equation 24, but for a large jump, the fit is very good. Consequently, even the extreme case of a sudden environmental change is well approximated in the moving optimum framework if (and thus the total number of adaptive steps) is moderately large.

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Figure 11.—

The distribution of adaptive substitutions in the sudden change scenario (i.e., the optimum immediately jumps to z_max). See Figure 10 for more details. In A, the solid line is the prediction from Equation 24. Parameters: haploid population with L = 10, K = 10³, u = 10⁻⁵, Θ = 2LKu = 0.2, . The distribution of new mutations is reflected exponential with .

The shape of the distribution:

Most adaptive substitutions are in the forward direction of the moving optimum (e.g., Figure 6). Although backward steps after a previous “overshooting” (Sato and Waxman 2008) are possible, they are very rare (Figure 2). The distribution of forward substitutions is unimodal: Most substitutions are of an intermediate effect, while both very small and very large steps are rare. Furthermore, the distribution is remarkably stable over the course of the adaptive process. That is, if the optimum of a perfectly adapted population starts moving, already the distribution of the first step is very similar to the equilibrium distribution after many steps. As a technical consequence, we can use the distribution of the first step (which is easier to derive) as a valid approximation for ϕ(α). Furthermore, all predictions about ϕ(α) also hold for short-term adaptation, where the movement of the optimum stops after a limited phenotypic distance. Our simulations show that the distribution of the first step approximates the distribution over an entire adaptive bout if the optimum changes by at least five times the average step size (Figure 10).

The selective sieve:

The distribution of adaptive substitutions can be decomposed into three factors: ϕ(α) = p(α)Ψ_s(α)Ψ_d(α) (Equation 20), where p(α) is the distribution of new mutations, and Ψ_s and Ψ_d are two sieve functions, which describe the effect of selection (Figure 3). The static sieve Ψ_s ∝ |α| reflects the fact that, independently of the ecological dynamics, alleles with small phenotypic effect have small fixation probabilities. As a consequence, ϕ(α) always approaches zero for α → 0. The dynamic sieve Ψ_d(α) captures the ecological and genetic dynamics. It cuts off mutations with large phenotypic effects when these would overshoot the moving optimum by more than the current phenotypic gap. The unimodal shape of ϕ(α) arises from the combined action of the two sieves.

The parameter γ:

To a large degree, the combined effect of the genetic and environmental dynamics on ϕ(α) depends on a single composite measure γ, which determines the properties of the dynamic sieve Ψ_d. In arbitrary units of the trait, γ is given by(Equation 19), where the distribution p(α) of new mutations enters through its standard deviation ω. The numerator describes the rate of environmental change, whereas the denominator can be understood as a measure of the adaptive potential of the population to follow the optimum.

Adaptive regimes:

Depending on the value of γ, we can distinguish two adaptive regimes, which form the endpoints of a continuum (Figure 4). The mutation-limited regime (large γ) is characterized by a large average gap between the population mean trait value and the optimum. As a consequence, the population follows the optimum with large steps, whose size is restricted only by the distribution of new mutations p(α). In the limit of γ → ∞, the dynamic sieve Ψ_d is “flat,” and the distribution of adaptive substitutions is independent of the ecological dynamics: ϕ(α) ∝ ∣α∣ p(α) (Equation 24). This limiting distribution corresponds to the one suggested by Kimura (1983) for Fisher's geometric model. In contrast, in the environmentally limited regime (small γ), the population tracks the optimum with only a narrow gap. Large mutations are usually not adaptive and are cut off by the dynamic sieve (which in this case is “steep”). Therefore, evolutionary change occurs in many small steps. For very small γ, ϕ(α) becomes independent of the shape of the distribution of new mutations and depends only on the selective sieve: ϕ(α) ∝ Ψ(α) (Equation 21).

The transition between the mutation and environmentally limited regimes occurs roughly over an interval γ ∈ [0.1, 10]. Biologically, γ is determined by the population size, the size of the mutational target, and the rate of environmental change (in mutational standard deviations per generation). In nature, these parameters can vary over many orders of magnitude. Small values of γ are expected, for example, for unicellular organisms with large population sizes and short generation times or under the gradual climatic shifts of geologic history. More generally, environmental limitation will almost necessarily play a role if the population is able to track a moving optimum for a long period of time. In our model, extinction risk is high already for γ ≥ 10 (simulation results not shown; cf. Lynch et al. 1991; Bürger and Lynch 1995; Nunney 2003). Purely mutation-limited adaptation is therefore possible only over short time spans.

Interaction effects:

As long as the frequency of adaptive substitutions is low or moderate, the results of the adaptive-walk approximation are fully confirmed by individual-based simulations. For strong “adaptive traffic,” however, we observe deviations due to interactions between cosegregating beneficial alleles (which are not accounted for in the adaptive-walk approximation). Interaction effects are most relevant for high mutation rates (Θ > 1), but are also favored by a fast moving optimum (which requires many substitutions per unit time). Therefore, the importance of interactions does not depend on γ and can be large in both the mutation-limited and the environmentally limited regime. We find two different types of interactions, which have opposite effects:

1. Linkage disequilibria between cosegregating alleles lead to Hill–Robertson (or clonal) interference, which induces a shift of ϕ(α) toward larger steps. This effect is well known from previous studies (e.g., Barton 1995). It can be strong if linkage is very tight (r < 0.01) or, equivalently, if the trait is affected by mutations at only a few loci (L < 10).

2. In contrast, interactions through negative epistasis for fitness (due to stabilizing selection) favor smaller alleles over larger ones. The reason is that multiple small alleles can have overlapping fixation times (because epistatic interactions between them are weak), whereas multiple large ones cannot.

Comparison to Kopp and Hermisson (2007, 2009):

The present article, which focuses on the long-term distribution of adaptive substitutions, complements two earlier studies on the moving optimum model (Kopp and Hermisson 2007, 2009), in which we investigated the fixation time of individual mutations and the order of substitutions over short bouts of adaptation. In the following, we summarize the picture arising from all three studies and highlight the similarities and differences between them.

All three articles stress the importance of the ecological timescale. They show that the rate of environmental change has a significant impact on the pattern of adaptive substitutions. If environmental change is sufficiently slow, the ecological timescale will influence or even dominate the adaptive process, causing the size of the most relevant mutations to decrease. In both the present study by and the recent study by Kopp and Hermisson (2009), the impact of the environment can be determined by a single composite parameter γ, which relates the rate of environmental change to the adaptive potential of the population. On the technical side, analytical results in all three articles build on a single-locus framework, in which interactions among segregating alleles are ignored. Simulations show that this approximation is valid in a large part of the biologically relevant parameter space.

However, there are also notable differences among the three articles. Most fundamentally, they address different questions. In Kopp and Hermisson (2007, 2009), we investigated the order of fixations and aimed at identifying the fastest substitution, that is, the substitution that initiates the adaptive response. In contrast, the present article considers the distribution of substitutions and, in particular, the most frequent substitution over a complete adaptive bout. Although the fastest and the most frequent substitution are often similar, they are not identical. The fastest substitution depends on the total time to fixation, whereas the most frequent one depends mainly on the (average) fixation probability. This difference has important consequences for the definition and demarcation of adaptive regimes.

When studying the order of substitutions (Kopp and Hermisson 2007, 2009), mutations with different effects can be seen as participating in a “race to fixation.” The outcome of this race depends on the total time to fixation, which has three components: the lag time until the allele becomes beneficial; the waiting time for the appearance of a successful mutation (which, in turn, depends on the mutation rate and the fixation probability); and the narrow sense fixation time, that is, the time the successful allele needs to sweep through the population. Accordingly, we distinguished three adaptive regimes with regard to the race-to-fixation problem (Kopp and Hermisson 2009): the environmentally limited regime, in which the total time to fixation is dominated by the lag time; the mutation-limited regime, in which it is dominated by the waiting time; and the fixation time-limited regime, in which it is dominated by the fixation time. Small mutations are either mutation or fixation time limited (i.e., they have long waiting or fixation times), whereas large mutations are environmentally limited (long lag times). The fastest mutations are those of intermediate size, which are located at the boundary to the environmentally limited regime.

Just as the fixation time in the previous articles can be decomposed into the lag time, the waiting time, and the fixation time, the distribution of adaptive substitutions can be decomposed into three factors: the distribution of new mutations p(α), the static sieve Ψ_s, and the dynamic sieve Ψ_d (Equation 20). In part, these decompositions are analogous. In particular, the lag time is implicit in the dynamic sieve and can be seen as the reason for why large mutations are cut off. However, there are also two important differences.

First, for the distribution of adaptive substitutions studied in the present article, the narrow sense fixation time is unimportant (and, consequently, is ignored in the adaptive-walk approximation). Once a mutation is picked up by selection and destined to go to fixation (which depends on the fixation probability), it does not matter how long the allele needs to actually reach frequency 1. In consequence, there is no fixation time-limited regime with respect to the distribution problem. (In fact, the fixation time still plays an indirect role by influencing the probability of interference between cosegregating alleles. With some right, one could therefore define an “interference regime” for the distribution problem to complement the fixation time-limited regime in the race-to-fixation problem. Note, however, that, in the race problem, the fixation time is relevant even in the absence of interaction effects.)

Second, in the present article, the distribution p(α) of new mutations enters as a (necessary) new player influencing the distribution problem. While, in the previous article, all mutations participating in the race to fixation were equipped with the same (recurrent) mutation rate θ, the traitwide mutation rate Θ in the present article is unevenly distributed across mutations with different effect. Since the waiting time for a successful mutation depends on both the fixation probability (i.e., the static sieve) and the mutation rate, there are now two ways in which the substitution rates of two mutations can differ as a result of differential “mutation limitation”: either due to differences in the fixation probability or due to differences in the mutation rate. The former possibility (differential fixation probabilities) is present in both articles and explains the pattern for small mutations (long fixation time, low substitution rate). The latter possibility (differential mutation rate) occurs only with a nonuniform distribution of new mutations and contributes to limiting the substitution rate of large mutations in the present article. However, for which kind of mutations (intermediate or large ones) this second kind of mutation limitation is more or less important than environmental limitation (i.e., the dynamic sieve) depends on the precise shape of p(α) (appendix b, Figure B1). In consequence, determining the limiting factor for the substitution rate of alleles with specific effect (as was done for the fixation time in Kopp and Hermisson 2009) can be complex (see appendix b). Instead, for the distribution problem, it proved more useful to define the adaptive regimes with respect to the whole adaptive process, that is, the entire distribution of substitutions. Indeed, the two regimes are best characterized via the limiting distributions given by Equations 21 and 24, respectively.

The nonuniform distribution of new mutations is also the reason why there is no direct correspondence between the summary parameters γ appearing in the two studies. In Kopp and Hermisson (2009), a parameter γ = v/(σθ) determines the effect of the fastest mutation α*, which defines the boundary to the environmentally limited regime. Unlike γ in the present study, it depends on the per-locus mutation rate θ, not on the per-trait mutation rate Θ. For nonuniform p(α), these two parameters cannot be straightforwardly translated into each other. Nevertheless, the two kinds of γ play essentially similar roles in describing the transition between mutation-limited and environmentally limited evolution.

Comparison to other models of adaptation and future directions:

Previous studies of adaptation to a single sudden change in the environment suggest that the distribution of adaptive substitutions should often be roughly exponential (Orr 1998, 2005a). In particular, an exponential distribution of adaptive substitutions has been obtained for both Fisher's geometric model (Orr 1998) and Gillespie's mutational landscape model (Orr 2002), although these two models are different in many aspects. For example, mutational effects are measured as phenotypic distances in Fisher's model and as fitness effects in the mutational landscape model. Yet, there are two main reasons why the two models show similar results (Orr 2005b): First, in both models, the distribution of new mutations is biased toward small mutations. Second, in both models, each adaptive step leads to a rescaling of the original situation: The population is closer to the optimum (in Fisher's model), or the likelihood of further beneficial mutations has decreased (in the mutational landscape model). As a consequence, the initial steps lead to sizable increases in fitness, whereas the later steps are mere “fine tuning.” It is this second point that is fundamentally different in the moving optimum model. For this reason, Orr (2005b) predicted that the distribution of adaptive substitutions in a moving optimum version of Fisher's geometric model should resemble that of the first step (originally calculated by Kimura), which is unimodal rather than exponential. This is, indeed, what was found in the present study. A similar unimodal distribution is also found for the fitness effects of adaptive substitutions (see appendix c, Figure C1).

However, a more important point seems to be what exactly determines the shape of the distribution of the first step. Here, we have shown that a “Kimura-type” distribution (Equation 24) arises only in the mutation-limited regime, where the mutational step size is limited by genetic constraints. In the environmentally limited regime, in contrast, the average mutational step size is no longer small relative to the distance to the (moving) optimum, and the distribution of adaptive substitutions is strongly influenced by the dynamic sieve. This leads to a significant decrease in average step size and also to a change in the shape of the distribution. In particular, the right-hand tail of ϕ(α) is proportional to exp(–Cα²) in the environmentally limited regime (see appendix b, Equation B1).

Interestingly, we usually do not find an exponential distribution of adaptive substitutions even in the case of a sudden and limited jump of the optimum (Figure 11). Indeed, if the jump is sufficiently large relative to the expected size of a single step, the overall distribution of step sizes is well approximated by the unimodal Kimura-type distribution (Equation 24). As seen in Figure 11C2, the mean size of subsequent steps declines only slightly, and the overall distribution of step sizes is almost identical to that predicted for the first step. The formal reason is that the distribution of new mutations p(α) introduces a fixed trait scale into the problem (the standard deviation ω) that does not rescale with the distance of the population to the optimum. As a consequence, the self-similarity arguments evoked by Orr (1998, 2005a) will generally not apply. Self-similarity is maintained, however, for the special case of a uniform distribution p(α), and, in this case, the distribution of the absolute values of step sizes approximately follows an exponential (in the limit of strong selection and weak mutation to avoid interference, results not shown). For the general case, Orr (1998) suggested to rescale the distribution of new mutations after each step with the distance to the optimum (i.e., to consider percentage changes with respect to the distance to the optimum). Obviously, this procedure reinstalls self-similarity. However, the assumption implies that the effect not only of substitutions, but also of new mutations decreases to zero as the population approaches the optimum. This seems to be biologically implausible at least for our model (but also for Fisher's geometric model considered by Orr 1998).

The current moving optimum model has been designed as a minimal model of adaptation to gradual environmental change. In its present form, it does not yet reflect the high dimensionality and nonlinearity of the genotype-to-phenotype-to-fitness map, which has rightfully been viewed as an essential characteristic of whole-organism adaptation (Fisher 1930; Orr 2005a,b). Future extensions should include multilocus phenomena, such as pleiotropy (as in Fisher's model) and epistasis (as in the mutational landscape model), and strive for a better understanding of the effects of linkage (as in clonal interference models). Another important topic is the role of standing genetic variation (Orr and Betancourt 2001; Hermisson and Pennings 2005; Barrett and Schluter 2008) in adaptation to gradual environmental change. Most generally, for a full understanding of adaptation, it will be necessary to take into account both the genetic complexities of real organisms and a realistic ecology.

So far, almost no data exist for testing the predictions of the moving optimum model. In particular, experimental evolution studies with microbes have almost exclusively employed constant selection regimes. Notable exceptions are provided by Collins and Bell (2004) and Perron et al. (2008), but a recent study by Collins and De Meaux (2009) is the first to (partly) resolve the evolutionary dynamics at the genetic level. For the future, we hope that more experiments will include gradual variation in environmental factors.

In microbial evolution experiments, it is often not possible to measure quantitative traits that mediate the relationship between genotype and fitness. Nevertheless, key assumptions and predictions of the moving model can also be tested in the absence of phenotypic data. For example, a basic prediction is that adaptation involves more substitutions in gradually changing environments than under constant selection. The number of substitutions might be estimated from DNA sequence data or by marker-based approaches as in Imhof and Schlötterer (2001). Furthermore, a hallmark of the moving optimum scenario is a reversal of fitness effects over time (i.e., selection favors first small and then large mutations). Thus, fitness assays should reveal evidence for antagonistic pleiotropy (fitness trade-offs) over the environments encountered during the experiment, just as previous studies have found antagonistic pleiotropy in environments not encountered by the population (Hughes et al. 2007a,b).

Conclusions:

In the present study and the two previous articles (Kopp and Hermisson 2007, 2009), we have shown that the adaptive process is strongly influenced by the dynamics of the selective environment. For adaptation to gradual and long-term environmental change, this influence is almost unavoidable. In particular, if the environment changes slowly, environmental factors are more important than genetic factors. We believe that this observation is robust also for other (more complex) models and may change or even reverse predictions obtained under the assumption of constant selection (e.g., small instead of large steps first, unimodal rather than exponential distribution of substitutions). Thus, our most general conclusion from this series of studies is also the most important one: Ecology matters for the genetic basis of adaptation.

Specific distributions of new mutations:

For specific distributions of new mutations, some of the integrals in the above equations can be evaluated explicitly (e.g., by using Mathematica). In particular, we can often obtain closed solutions for the function g(δ) (Equation A5).