THE EFFECTS OF POPULATION BOTTLENECKS ON CLONAL INTERFERENCE, AND THE ADAPTATION EFFECTIVE POPULATION SIZE

Authors


Abstract

Clonal interference refers to the competition that arises in asexual populations when multiple beneficial mutations segregate simultaneously. A large body of theoretical and experimental work now addresses this issue. Although much of the experimental work is performed in populations that grow exponentially between periodic population bottlenecks, the theoretical work to date has addressed only populations of a constant size. We derive an analytical approximation for the rate of adaptation in the presence of both clonal interference and bottlenecks, and compare this prediction to the results of an individual-based simulation, showing excellent agreement in the parameter regime in which clonal interference prevails. We also derive an appropriate definition for the effective population size for adaptive evolution experiments in the presence of population bottlenecks. This “adaptation effective population size” allows for a good approximation of the expected rate of adaptation, either in the strong-selection weak-mutation regime, or when clonal interference comes into play. In the multiple mutation regime, when the product of the population size and mutation rate is extremely large, these results no longer hold.

It is well established that adaptation occurs through the appearance and eventual fixation of advantageous mutations in natural populations. In the simplest scenario, beneficial mutations are very rare and the rate of adaptation is bounded by their availability. In this case fixation events alternate with periods during which the population waits for the appearance of the next advantageous mutation, a scenario known as periodic selection (Atwood et al. 1951), the “successional-mutation regime” (Desai et al. 2007), or the strong-selection weak-mutation regime (Gillespie 1983; Gillespie 1984).

With larger population sizes or higher mutation rates, beneficial mutations arising in distinct lineages may compete for fixation. Such competition among beneficial alleles, “clonal interference,” increases the expected time to fixation, and consequently slows the overall rate of adaptation (Hill and Robertson 1966; Gerrish and Lenski 1998). At even higher mutation rates or population sizes, multiple beneficial mutations are likely to occur in the same segregating lineage, while it is en route to fixation; this is the “multiple mutation regime” of Desai and Fisher (Desai et al. 2007; Brunet et al. 2008; Rouzine et al. 2008).

Interest in studying adaptation in natural and laboratory populations has been recently renewed, due in part to recent efforts in estimating the mutation rate to novel beneficial alleles in microbial populations (Joseph and Hall 2004; Desai et al. 2007; Eyre-Walker and Keightley 2007; Perfeito et al. 2007a). These investigations indicate a much larger influx of beneficial mutations than previously predicted (Miralles et al. 1999; Imhof and Schlotterer 2001; Rozen et al. 2002), suggesting that clonal interference plays a critical role. Several mechanisms have been considered to affect the strength of clonal interference, such as, rates of beneficial and deleterious mutations (Orr 2000; Campos and de Oliveira 2004; de Oliveira and Campos 2004; Wilke 2004), population structure (Gordo and Campos 2006; Habets et al. 2007; Perfeito et al. 2007b), and environmental conditions (Pepin and Wichman 2008).

Population bottlenecks, sudden and severe reductions in the population size, are an inherent feature of many of the experimental protocols used in these microbial evolution experiments. Periodic bottlenecks are necessitated by the often rapid growth rates of the microbial populations under study. Although the effects of population bottlenecks on the fixation probability of beneficial mutations have been described in some detail (Wahl and Gerrish 2001; Wahl et al. 2002), the effects of population bottlenecks on clonal interference have not yet been examined.

To account for the changing population size in these protocols, a number of recent experimental papers (Gerstein et al. 2006; Maughan et al. 2006; Hill and Otto 2007; Masel and Maughan 2007; Perfeito et al. 2007a) have made use of an effective population size, Ne, proposed for populations that grow exponentially between repeated bottlenecks (Wahl and Gerrish 2001). This value of Ne was determined by considering the fixation probabilities of beneficial mutations that first arise at different times between population bottlenecks. However, effective population sizes must be defined with respect to a specific stochastic process or quantitative behavior; when the overall influx (occurrence and fixation) of beneficial mutations is the quantity of interest, this previously derived Ne may not be appropriate.

In the sections that follow, we describe a mathematical approximation for the rate of adaptation in the presence of both clonal interference and bottlenecks. We compare these analytical predictions to the results of an individual-based simulation of clonal interference, in the presence of population bottlenecks, based on a Wright–Fisher model for population growth. We demonstrate that when the product of the population size and mutation rate is small enough that mutations do not selectively sweep in groups, previous theory for clonal interference holds, as long as bottlenecks are taken into account in determining the fixation probability.

Finally, we derive an appropriate definition for the effective population size for adaptive evolution experiments. This “adaptation effective population size” allows for prediction of the rate of adaptation in the presence of population bottlenecks, either in the strong-selection weak-mutation regime, or when clonal interference comes into play. In the multiple mutation regime, when the product of the population size and mutation rate is extremely large, our theory underestimates the rate of adaptation (Desai et al. 2007; Brunet et al. 2008; Rouzine et al. 2008).

Methods

WRIGHT–FISHER SIMULATION OF CLONAL INTERFERENCE

We simulate a finite population of N0 individuals that experience a growth phase for τ discrete generations. Upon reaching size Nf the population is subjected to a bottleneck procedure, in which N0 individuals are chosen at random to form the next founding population. During the growth phase the population evolves according to the standard Wright–Fisher model with nonoverlapping generations, where the individuals in generation t+ 1 are generated by sampling with replacement from the individuals in generation t. The number of offspring of each individual in generation t is Poisson-distributed, with a mean proportional to the individual's fitness.

The model assumes that adaptive mutations take place at rate ub per individual per generation. The selective effect of each mutation, sb, is sampled from an exponential distribution according to inline image where 1/α corresponds to the mean value of sb.

In our simulations, we treat a wide parameter range, encompassing successional mutations, clonal interference, and multiple mutations. In the latter regime, by definition, it is possible that an individual may carry more than one segregating beneficial mutation. In this case, the fitness of the individual, ωi, is a function of the number of advantageous mutations that individual carries, kb, and their selective effects

image(1)

where sb,j is the selective effect of the jth mutation, and we have assumed a multiplicative fitness landscape. As described in greater detail below, however, our analytical work treats the clonal interference regime, and thus this assumption does not come into play.

Our simulations begin with N0 wild-type individuals, which replicate in each generation, possibly mutating, and face population bottlenecks every τ generations. We keep track of the fate of each mutation produced along the evolutionary trajectory, and then at the end of each run we count the number of those mutations that have reached fixation. We let the population evolve for 2000 bottlenecks, and our results are averaged over 50 independent replicates.

MATHEMATICAL MODEL OF ADAPTATION RATE

Following Wahl and Gerrish (2001), we use a continuous time approximation to the Wright–Fisher model described above. Assuming that the wild-type population grows exponentially at rate r during the growth phase, the probability that a new beneficial mutation ultimately survives bottlenecks can be approximated as 2rsbτert, where t is the time, during the growth phase, at which the new mutation first arises. Using this result together with the Gerrish–Lenski theory of clonal interference (Gerrish and Lenski 1998), we compute the expected rate at which new beneficial mutations arrive, at time t, which will ultimately reach fixation

image(2)

Here inline image is the derivative of N with respect to time (in this case, inline image), and ub is the mutation rate, per replication, for beneficial mutations. Thus inline image is the rate at which new beneficial mutations are expected to arise. The term inline image is the probability density of selective effects sb, which has been shown to be roughly exponential over small sb for a large family of underlying fitness distributions (Orr 2003). The term 2rsbτert is the probability that the mutation of interest is not lost during bottlenecks. Finally, the term eI gives the probability that the mutation of interest is not lost by clonal interference. Here I is the number of interfering mutations that are expected to arise while the mutation of interest is en route to fixation. We define “interfering mutations” as all mutations with a larger selective advantage than the mutation of interest and that also survive bottlenecks, and we assume in this mathematical approximation (but not in the simulations) that the occurrence of any such mutations will lead to the extinction of the mutation of interest. Specifically, assuming that the number of interfering mutations is Poisson-distributed, the term eI corresponds to the probability that exactly zero interfering mutations occur before the mutation of interest is fixed.

Simplifying, we see directly that

image(3)

which does not depend on t, the time during the growth phase at which the mutation first appears. Thus, E[Kb] should approximate the overall rate of fixation of beneficial mutations.

To estimate I, we need the rate at which interfering mutations occur, the probability that these interfering mutations are not themselves lost to bottlenecks, and the expected time before the mutation of interest reaches fixation, Tfix. Again according to Wahl and Gerrish, one can estimate the number of mutations which occur at time t with benefit effect larger than sb, and which will not be lost to bottlenecks, as

image(4)

Again, we note that this equation is independent of t, and so the rate at which superior (s > sb) mutations occur is roughly constant over time.

Although equation (4) gives the overall rate at which superior mutations occur, these mutations are only interfering if they occur in a wild-type individual; if superior mutations occur in an individual carrying the mutation of interest, they will not interfere but will in fact accelerate fixation. Because the rate λ is independent of t, we can multiply λ by F, the average fraction of the population that consists of wild-type individuals, averaged over the entire time interval during which the mutation of interest is en route to fixation. Thus we have I≈λ(sb) FTfix.

DIFFUSION APPROXIMATION

To estimate both F and Tfix, we use a standard diffusion approximation. The key novelty in our approach is that we consider a “time step” in the diffusion approximation to be a complete cycle of population growth and sampling. This is justified because the bottlenecks reduce the wild-type population to its initial size, implying that the change in frequency of a beneficial mutation from the beginning of one growth phase to the next will be small, as long as s is not too large.

Consider a beneficial mutation that has frequency p at the beginning of a growth phase, such that the frequency of the wild-type is q= 1 −p. Let p1 be the expected frequency of the mutant after one growth phase (that is, after τ generations of growth), immediately before the first bottleneck. Likewise, let p+1 be the expected frequency of the mutant immediately after the first bottleneck.

During growth, each of the pN0 mutant individuals has on average er(1 +sb) offspring per generation, for τ generations. The entire population is then sampled at rate D=erτ. Thus p+1 is given by the expected number of mutant individuals over the expected total population size after growth and sampling

image(5)

To get the expected value of the change in frequency from the beginning of one growth phase to the next, we compute

image

because the denominator is close to one for all values of p, as long as neither sb nor τ are too large.

The next step in the diffusion approximation is to estimate the variance in the change in frequency (again, from the beginning of one growth phase to the beginning of the next). Here we make the simplifying assumption that most of the variance will be due to the sampling process, not the stochastic growth of the mutant strain. We thus assume that the frequency of the mutant strain just before the bottleneck is equal to its expected value at that time, p1

image

The variance of the change in frequency is then given by the variance of a binomial distribution, in which N0 individuals are chosen, to form the next founding population, and the probability that a mutant is chosen is given by p1. This yields the variance

image

M and V can then be used in the standard diffusion approximation for the fixation time (see Gale 1990 p. 256). We use this to find the probability density for the sojourn time, that is the total time spent by the mutant population at frequency x, while en route to fixation. If the initial frequency of the mutant strain px, this density is given by

image

where a= ((1 +sb)τ− 1)/(1 +sb)τ, or asbτ when sb is small. Here we have assumed that N0 is sufficiently large that inline image is negligible.

We note that our final approximation for T*(p, x) is symmetrical in the terms x and 1 −x; despite the selective advantage, the time spent by a beneficial mutation at frequency x en route to fixation is about the same as the time spent at frequency 1 −x. This implies that the average frequency of the mutant en route to fixation is 0.5, yielding F= 0.5.

To estimate the mean time to fixation, we integrate T*(p, x)

image

The right-hand side is multiplied by τ to convert the time units from time steps (one growth and sampling phase) to generations.

In many of the Results to follow, we compute this integral numerically to estimate Tfix. However, a closed-form approximation will be necessary in developing the effective population sizes in the following section. To obtain this, we first note that because of symmetry, we can compute the integral above from x= 1/N0 to x= 0.5 and multiply by two. In this range, inline image is small, and following Gale (1990), we write

image

The first term on the right, evaluated at a= 1/N0 and b= 0.5 yields a−1 (ln (N0) − 1/N0). The second term is negligible except when x is very small, and in this range (1 −x) is close to unity. Thus we can use the standard result for inline image (Abramowitz and Stegun 1964) to find that the second term is approximately inline image. Subtracting the second term from the first, doubling the result, and dropping negligible terms, we arrive at

image

Technically, we should also add the time spent on average at frequency x= 1 − 1/N0, which is two generations (Gale 1990), changing the final term above to -2. However we see immediately that for large populations, a single term will dominate, leaving us with

image(6)

because asbτ.

ADAPTATION EFFECTIVE POPULATION SIZE

An important concept in population genetics is the effective population size, Ne. This corresponds to the size of an ideal Wright–Fisher population that would have the same quantitative behavior as the more complex population in question. The behavior of interest might be for example the variance in offspring number (variance effective population size), or the probability that two alleles chosen at random are from the same parent (inbreeding effective population size).

To define an “adaptation effective population size,” we use the following equation to estimate the rate at which mutations of selective advantage sb will occur and fix

image(7)

Here Neub should give the overall rate at which beneficial mutations occur. The term P(sb) gives the probability density for mutations of effect sb; typically, we take P(s) =αe−αs, but any other distribution may be assumed. The fixation probability is assumed to be 2sb. This equation holds for a large population of constant size with Poisson-distributed offspring, in the strong-selection weak-mutation regime; in this simple case Ne=N, the census size.

Population bottlenecks: Now consider a wild-type population that begins at size N0, and grows exponentially at rate r for τ time units, reaching size Nf=N0erτ. The population then undergoes a bottleneck, in which fraction D=erτ of the final population size is sampled to form the next founding population. In this case we assume that only the new individuals in the population in each generation can mutate, such that the number of new mutations is given by (dN/dtub rather than N ub. Using the approximation 2sbrτert for the fixation probability when mutations are rare, we write

image

(note that we have divided by τ to obtain κB in units of fixation events per generation). Comparing with equation (7), it is clear that for populations that grow between periodic bottlenecks, the effective population size for the influx of beneficial mutations is

image(8)

In other words, the overall rate at which beneficial mutations are expected to occur and survive bottlenecks will be the same as in a constant population with size Ne and fixation probability 2sb.

Clonal Interference: We now consider a population of constant size, but in this case relax the assumption of strong-selection weak-mutation. We relax this assumption just enough to allow for competing beneficial mutations to interfere with fixation, but not enough that multiple segregating beneficial mutations occur at high frequency on the mutant background (the multiple mutation regime). In this case, we write

image(9)

where IC is the number of interfering mutations, that is, the number of mutations with selective advantage greater than sb, which occur while the mutation of interest is en route to fixation and survive drift. Following Gerrish and Lenski, we have for example

image(10)

if we take inline image and 2sb as the fixation probability. We note that IC depends sensitively on sb, and thus it would be impossible to define an effective population size that would reconcile equation (9) with equation (7).

Population bottlenecks and Clonal Interference: Finally, we consider the case of clonal interference described above, in a population that grows between bottlenecks. In this case, we have

image(11)

Comparing with equation (9), it appears that Ne=r2τN0 may also be the appropriate effective population size in this case. However, in this case we must also consider

image

which is not the same as IC. We take F= 0.5, and consider very large population sizes. Taking Tfix≈ 2 ln(N0)/sb, we find

image(12)

Comparing this expression for IBC with equation (10), we see that clonal interference in the presence of periodic bottlenecks should have similar effects as clonal interference in a constant population size of

image(13)

This will be approximately true as long as ln(N0) ≈ ln(r2τN0) = 2 ln(r) + ln(τ) + ln(N0), that is, as long as ln(N0) is much larger than ln(r) and ln(τ).

INCLUDING DELETERIOUS MUTATIONS

In asexual populations, deleterious mutations are expected to produce a severe reduction in the rate of adaptation (Felsenstein 1974; Charlesworth 1994; Peck 1994; Barton 1995; Orr 2000; Johnson and Barton 2002; Campos and de Oliveira 2004; Bachtrog and Gordo 2004; Wilke 2004), because the occurrence of deleterious mutations leads to a reduction of the effective population size (Fisher 1930). Adaptation rates in asexual populations have been predicted, in the presence of both beneficial and deleterious alleles, by Orr (2000), with elegant asymptotic results provided by Wilke (2004). In the simplest scenario, only those advantageous mutations arising in the most adapted class of individuals contribute to the adaptive process. In large populations, this implies that only beneficial mutations occurring in a genetic background free of deleterious mutations have a nonzero chance of reaching fixation, i.e., sdsb, where sd is the selective effect of the deleterious mutation.

Previous authors have also assumed that beneficial mutations occur in a heterogeneous genetic background that is at the mutation–selection balance, in which case the frequencies of the classes of individuals carrying kd deleterious mutations are Poisson-distributed with parameter Ud/sd, where Ud is the mutation rate to deleterious mutations (Haigh 1978). This requires the population to be sufficiently large such that there is no loss of the most fit class of individuals due to stochastic effects.

More recently, Bachtrog and Gordo (2004) have relaxed these assumptions to extend their analysis to situations in which populations are not necessarily at mutation–selection equilibrium. Specifically, they studied how Muller's ratchet, the continuous loss of the best-adapted individuals due to stochastic events (Muller 1964), affects the adaptive process. They have shown that in this case the rate of fixation of advantageous mutations can be higher than expected for background selection because the instability of the class of best genotypes can promote the fixation of favorable mutations arising in more loaded classes of mutants. They have also studied the case, in the presence or absence of Muller's ratchet, in which sb > sd, such that beneficial mutations arising in mutant classes other than the best-adapted class may reach fixation.

The existence of periodic bottlenecks further complicates these issues; when the population size at the bottleneck is small, the time between bottlenecks may not be long enough to ensure that the population reaches mutation–selection equilibrium before the next beneficial mutation occurs (Gordo and Dionisio 2005).

Although a full exploration of the effects of population bottlenecks in each of these parameter regimes would be beyond the scope of this manuscript, here we present some preliminary results under assumptions that will be relevant to many experimental protocols. Specifically, we assume that the population size, throughout the growth phase, is sufficiently large that Muller's ratchet is not operating, and that the mutation–selection balance obtains, even immediately after a bottleneck. As before, we assume that beneficial effects are exponentially distributed with mean 1/α, but for mathematical tractability we assume that the selective effect of deleterious alleles, sd is fixed. We then treat the case in which sd≫ 1/α. Because sb is drawn from a probability distribution, this means that most beneficial alleles will only fix if they occur in an individual that is free from deleterious alleles. However we note that in our simulations, rare beneficial alleles that confer large values of sb can compensate for one or more deleterious alleles.

As in the case of beneficial mutations, we assume that deleterious mutations take place at a constant rate Ud, and each deleterious mutation reduces fitness by a constant factor (1 −sd). Considering the multiplicative fitness landscape, the fitness of an individual carrying kb advantageous mutations and kd deleterious mutations is now calculated as

image(14)

(Once again we note that the possibility of more than one segregating beneficial mutation occurring in the same individual is relevant only to our simulation results.)

Under the assumptions above, we can determine the substitution rate by imposing the approximation that beneficial mutations will only reach fixation if they first occur in individuals that carry no deleterious mutations. Thus equation (3) can be rewritten as

image(15)

where f0 is the fraction of individuals that are mutation-free, given by exp (−Ud/sd) (Haigh 1978), and If is the expected number of interfering mutations, assuming that these must also occur in mutation-free individuals.

Similarly, one can estimate the number of mutations that occur at time t with benefit effect larger than sb, which arise in an individual free of deleterious alleles, and which will not be lost to bottlenecks, as

image(16)

As before, we now need to estimate If≈λf(sb) F Tfix. But in this case Tfix must also take into account the occurrence of deleterious alleles.

We proceed as follows: in the large population at mutation–selection balance that we consider, the mean population fitness in the absence of beneficial mutations is inline image (Haigh 1978), whereas the fitness of the beneficial mutation of interest is (1 +sb). It is this change in the mean population fitness, which will (slightly) alter the time to fixation; the presence of deleterious alleles has no other effect on F or Tfix.

During growth, each of the pN0 individuals carrying the beneficial mutation has on average er(1 +sb) offspring per generation, for τ generations. Because the mean fitness of the rest of the population is inline image, each of the individuals that do not carry the beneficial mutation has on average inline image offspring per generation. The entire population is then sampled at rate inline image. Thus p+1 is given by the expected number of mutant individuals over the expected total population size after growth and sampling

image

When Ud is relatively small, we have inline image. Thus, comparing with equation (5), we see that our previous diffusion approximations should hold, but sb should be replaced by sb+Ud in computing Tfix. This makes good intuitive sense because the mean population fitness has been reduced by approximately Ud.

Results and Discussion

Figure 1 illustrates the close agreement between our simulation results (filled circles; error bars not visible), and the numerical approximation of E[Kb] described in the previous section (solid black lines), for two population sizes (upper and lower panels). For comparison, we also plot the simulated fixation rate, Kb, for a population of constant size experiencing clonal interference; in these cases the population size was fixed at either N0 (diamonds), or Nf (squares). The dashed-lines in the figure correspond to the analytical prediction for E[Kb] in a population of constant size, as exactly derived by Gerrish and Lenski (1998). In a constant-sized population

image(17)

and

image(18)
Figure 1.

Rate of fixation, Kb, as a function of mutation rate ub. The parameter values are Nf= 65536, α= 20, r= ln 2; (A) N0= 512 and (B) N0= 2048. The datapoints correspond to the simulation results averaged over 50 replicates, whereas the solid black line is the theoretical prediction according to equation (2). For comparison, simulation results in populations of constant size N0 (diamonds) and Nf (squares) are also shown. The dashed-lines give analytical predictions for these cases (see text for details). The gray line gives the analytical approximation based on the adaptation effective population size.

At small and intermediate Ub the rate of fixation Kb for the population subjected to periodic bottlenecks lies in between the results for the constant population sizes N0 and Nf. When the novel advantageous mutations are not frequent, most mutations occur in isolation. In this case bottlenecks drastically reduce E[Kb], bringing the expected adaptation rate much closer to that predicted for N0 than for Nf. As Ub is increased and NfUb becomes larger than one, clonal interference becomes prevalent and the competition between mutations in distinct lineages slows down the adaptation rate. This effect is most pronounced for the larger population size, and population bottlenecks reduce this competition, allowing the adaptation rate to increase in parallel with the smaller population size over a range of ub. At very large Ub the simulated fixation rates for all population sizes coalesce.

The gray lines in Figure 1 correspond to the numerical solution of equation (17), substituting the effective population size Ne=N0r2τ for N. Again, we note excellent agreement between simulation results and theory in the isolated mutation regime, and when clonal interference dominates. This agreement verifies the use of the adaptation effective population size when the rate of influx of beneficial mutations is the quantity of interest. However, at extremely large ub this theoretical result underestimates the simulated fixation rate, as we also observe in the theoretical lines for the constant-sized populations. This underestimate occurs when novel mutations are likely to occur on the background of a currently segregating beneficial mutation; our analytical predictions no longer hold in the multiple mutation regime.

The mean selective effect of mutations that ultimately fix is also of interest. Figure 2 illustrates the mean selective effect for all the mutations that fixed in our simulation studies of populations subjected to periodic bottlenecks. We see an initial increase in the mean selective effect as the mutation rate increases; this is a clear signature of clonal interference, which reduces the fixation probability of small effect sb mutations. On the extreme right of the figure, the mean selective effect decreases at very high mutation rates; in this regime, many mutations of smaller effect hitchhike to fixation, reducing the mean value of sfix, which corroborates the existence of the multiple mutation regime. Moreover, we notice that more severe bottlenecks reduce the mean selective effect of those mutations that ultimately fix. This result is in accordance with empirical results (Perfeito et al. 2007a).

Figure 2.

Mean selective effect of those mutations that have reached fixation, inline image, as a function of mutation rate ub. The parameter values are Nf= 65536, α= 20, r= ln 2, and N0= 512 (filled circles) and N0= 2048 (empty circles).

Figure 3 displays the distribution of the selective effects of those mutations that ultimately reached fixation. We show the distribution for different values of mutation rate ub. In the figure we compare our simulation results with our theoretical prediction. Thus, the solid lines are the numerical solution of the following distribution:

image(19)

and IBC is given by equation (12). For low and intermediate values of the mutation rate (ub= 1 × 10−5 and 1 × 10−4, respectively), we notice a very good agreement between theory and simulation. In these two cases, the selective effects of fixed mutations exhibit a skewed, unimodal distribution, in accordance with empirical results (Rozen et al. 2002; Perfeito et al. 2007a). At very small ub, the coexistence among distinct advantageous mutations is unlikely, and small-effect mutations are largely lost due to drift and population bottlenecks. On the other hand, large-effect mutations have a much improved chance of reaching fixation but these are not very common. When ub= 1 × 10−4 we observe that the distribution is shifted toward higher values of s because clonal interference decreases the likelihood of fixation of mutations with small and intermediate effects. In panel (C) of the same figure we check that the theoretical prediction, denoted by equation (19), fails to predict the simulation results. In this case, small-effect mutations have an enhanced likelihood of fixation compared to the previous cases, which again confirms our claim that the multiple mutation regime is responsible for the decay in sfix seen at high mutation rates in Figure 2.

Figure 3.

Probability distribution, P(sfix), of selective effects of those mutations that have reached fixation. The parameter values are Nf= 65536, α= 20, r= ln 2, and N0= 512 and mutation rate ub= 1 × 10−5 (A), ub= 1 × 10−4 (B) and ub= 1 × 10−3 (C). The datapoints are our simulation results, whereas the solid line is the theoretical prediction.

Finally, in Figure 4 we can see the effect of the deleterious mutations on the rate of fixation E[kb]. We compare the simulation results with the analytical expression equation (15), taking into account the change in our estimated Tfix. In these simulations we use the deleterious mutation rate Ud= 0.1. The upper curve corresponds to sd= 0.1 and α= 20, the middle curve has sd= 0.1 and α= 50, whereas the lower curve corresponds to sd= 0.05 and α= 50. In all cases sd > 1/α; in the middle curve in particular, a quick calculation reveals that the chance that sb > sd is less than 0.01. The assumption that sd always exceeds sb is violated for about 10% of beneficial mutations in the upper and lower curves. Nonetheless, we obtain a good agreement between simulations and theory. In comparing these cases with the corresponding case when Ud= 0, we observe that deleterious mutations lead to a reduction in the rate of substitution of favorable mutations, but this effect becomes less pronounced as Ub increases. At large Ub, where clonal interference operates, the curves collapse and there is little distinction between the two situations. This clearly indicates that the reduction in the rate of fixation due to a smaller effective population size is directly compensated by a concomitant reduction in the strength of clonal interference.

Figure 4.

Rate of fixation, Kb, as a function of mutation rate ub, in the presence of deleterious mutations. The parameter values are N0= 512, Nf= 65536, r= ln 2, and Ud= 0.1. Simulation results for sd= 0.1 and α= 20 (filled circles), sd= .1 and α= 50 (triangles) and sd= 0.05 and α= 50 (open circles) are shown. The solid lines give analytical predictions for these cases according to equation (15).

The adaptation effective population size we derive here, Ne=r2τN0, differs by a factor of r, the wild-type population growth rate, from an effective population size Ne=rτN0 previously derived for populations that grow exponentially between periodic bottlenecks (Wahl and Gerrish 2001). The previously derived Ne was found by assuming that the fixation probability should have the form 2sbNe/Nt, where Nt is the population size at the time of occurrence of the mutation (Kimura 1964). This Ne is thus appropriate when comparing the fixation probabilities of beneficial mutations that first arise at different times during the growth phase.

In contrast, the effective population size Ne=r2τN0 has been derived by assuming that the fixation rate of beneficial mutations should have the form given in equation (7). This adaptation effective population size is appropriate when comparing the rate of adaptation in populations subjected to periodic bottlenecks, in either the strong-selection weak-mutation or clonal interference regime.

Associate Editor: L. Meyers

ACKNOWLEDGMENTS

This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), program PRONEX/MCT-CNPq-FACEPE and the Natural Sciences and Engineering Research Council of Canada.

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