The rate at which a population adapts to its environment is a cornerstone of evolutionary theory, and recent experimental advances in microbial populations have renewed interest in predicting and testing this rate. Efforts to understand the adaptation rate theoretically are complicated by high mutation rates, to both beneficial and deleterious mutations, and by the fact that beneficial mutations compete with each other in asexual populations (clonal interference). Testable predictions must also include the effects of population bottlenecks, repeated reductions in population size imposed by the experimental protocol. In this contribution, we integrate previous work that addresses each of these issues, developing an overall prediction for the adaptation rate that includes: beneficial mutations with probabilistically distributed effects, deleterious mutations of arbitrary effect, population bottlenecks, and clonal interference.
Understanding the evolutionary forces that dictate the evolution of natural populations has long been an issue of great interest both within and outside the scientific community. A critical step in this direction was taken at the beginning of the 20th century with the foundation of the modern theory of evolution, which states that evolution is the result of the interplay between two antagonistic mechanisms: natural selection and sources of genetic variation. The advent of this theory enabled a rapid development in the field of theoretical population genetics (Haldane 1927; Fisher 1930; Wright 1931). More recently, experimental advances have made long-term evolution experiments feasible, particularly in microorganisms (Lenski et al. 1991; Lenski and Travisano 1994). Today, there is a vibrant experimental evolution community and most of our recent achievements in the field come from empirical observations in experiments involving microorganisms such as bacteria, viruses, fungi and yeast (Elena and Lenski 2003).
The experimental advances themselves have presented further challenges to the development of quantitative, predictive models. Because of the rapid growth rate of many microbial populations, most microbial evolution experiments involve population bottlenecks, sudden and severe reductions in the population size, which are imposed periodically as a fresh medium is inoculated with a sample from the evolving population. Typically, an effective population size, Ne, is used to modify theoretical predictions for a changing population size. However, effective population sizes must be defined with respect to a specific stochastic process or quantitative behavior. Recently, we derived the “adaptation effective population size,” which is the appropriate definition of an effective population size when the process of interest is the adaptation rate (Campos and Wahl 2009). The adaptation effective population size is valid for populations which are subject to periodic bottlenecks, and also when a large influx of beneficial mutations occur, such that the simplest scenario of SSWM (Gillespie 1983) is no longer valid. This enabled us to accurately predict the rate of substitution of advantageous mutations in the clonal interference regime, in the absence of deleterious mutations.
Clearly, a complete picture of adaptation should also include deleterious mutations, which are certainly abundant (Keightley 1994; Drake and Holland 1999), and which play a remarkable role in adaptation because their presence greatly influences the fate of beneficial mutations, and consequently affects the strength of clonal interference (Felsenstein 1974; Charlesworth 1994; Orr 2000; Campos and de Oliveira 2004; de Oliveira and Campos 2004; Wilke 2004; Bachtrog and Gordo 2004). It is well established that deleterious mutations cause a severe reduction in the adaptation rate, as a consequence of reducing the effective population size. The simplest situation corresponds to the case in which only beneficial mutations that occur in individuals that are mutation-free contribute to the adaptive process; this is valid when the effect of each deleterious mutation on fitness, sd, is larger than the benefit sb brought about an advantageous mutation. However, deleterious mutations of small effect are probably more likely (Keightley 1994; Loewe and Charlesworth 2006), and beneficial mutations of extremely large effect have been reported (Bull et al. 2000).
In this contribution, we predict the adaptation rate for asexual populations, including deleterious mutations of arbitrary magnitude, and further including the possibility of both clonal interference and population bottlenecks. We compare simulation results for the rate of fixation of favorable mutations with theoretical predictions obtained through two different approaches.
We consider a finite population of haploid individuals that grows exponentially for a period of length τ. In our Wright–Fisher simulation of this growth phase, the population size N(t) changes in discrete, nonoverlapping generations according to N(t) =N0ert (t= 0, 1, … , τ). Individuals in time t contribute to form the next generation t+ 1 with probability proportional to their fitness.
After this growth, the population is subjected to a bottleneck process, where N0 individuals are randomly chosen from the Nf=N0erτ individuals. This process is then repeated. During reproduction, individuals can acquire new mutations (beneficial and/or deleterious) at constant rates Ub (beneficial) and Ud (deleterious), and the number of beneficial and deleterious mutations that a given individual acquires in a single generation is Poisson distributed. Moreover, we assume that the effects of the deleterious mutations are independent and constant, that is, each deleterious mutation reduces fitness by a factor (1 −sd), whereas beneficial mutations are assumed to have independent effects but the magnitude sb of each is drawn from a probability distribution P(sb). In our simulation results, we have used an exponential distribution with mean 1/α, that is, , but the use of other distributions would be straightforward (see Rokyta et al. 2008). The assumption for the exponential distribution of the selective effects of adaptive mutations is based on previous arguments of extreme value theory (Gillespie 1991; Orr 2003), and also has some experimental support (Rozen et al. 2002). Thus, in the model an individual carrying Nb beneficial and Nd deleterious mutations has fitness
We initiate our simulations with N0 mutation-free individuals. Before introducing beneficial mutations, the population first evolves for teq= 100 bottlenecks in such that an equilibrium regime, the mutation-selection balance, is reached. After this period, advantageous mutations are enabled and we keep track of all beneficial mutations during the evolutionary trajectory. In total, we let the population evolve for 5000 bottlenecks, and the simulation results are averaged over 50 independent runs. At the end of each simulation, we count the number the mutations that have reached fixation.
These simulations were replicated using two independently written algorithms. In one program, we track each of up to Nf individuals, storing which beneficial and how many deleterious mutations the individual carries. In this algorithm, we model the following life cycle: reproduction, mutation, and selection, where selection is applied by choosing individuals proportional to fitness to survive and form the next generation. In a second, complementary program, we track the number of genetically identical individuals in each “class,” where a class is identified by Nd, Nb, and a set of Nb values of sb,j. In this algorithm, the life cycle is: reproduction proportional to fitness, followed by mutation. The first algorithm performs very well at high mutation rates but becomes inefficient when the population size is very large; the second algorithm performs well for large population sizes, but becomes inefficient when the mutation rate, and thus the number of distinct classes, becomes large. Over the parameter range where both algorithms are applicable, the results from the two programs are indistinguishable, offering good evidence for the integrity of the code in both cases.
Analytical Methods and Results
In the strong-selection weak-mutation (SSWM) model, beneficial mutations are assumed to have strong effects, and additionally to be so rare that the chance of coexistence of advantageous mutations in distinct lineages in a population is negligible. In this very simple model, the rate at which advantageous mutations fix, Kb, in a population of fixed size N, is given by
where Nub is the number of new beneficial mutations per generation, and π(sb) is the chance that a favorable mutation of effect sb escapes extinction when rare, usually approximated as 2sb when sb is small (Haldane 1927).
The above expression overestimates Kb when rates of favorable mutations are large, because now beneficial mutations in distinct lineages can coexist and compete for fixation (clonal interference). This competition occurs when novel beneficial mutations occur, and escape extinction when rare, faster than they reach fixation, or when Nubπ(sb)Tfix > 1, where Tfix is the fixation time. The transition to the MM regime (Brunet et al. 2008; Rouzine et al. 2008b) occurs when Kb≥ 1/Tfix. Because both Tfix and π depend sensitively on sb, it is difficult to compute where these transitions occur for a distribution of beneficial effects, especially since the distribution of mutations that ultimately fix is quite different from P(sb) (Campos and Wahl 2009). In the sections that follow, we derive I, the mean number of interfering mutations, and Ne, the adaptation effective population size. We then use these concepts to give an alternate approximation of the range of the clonal interference regime.
In equation (2) the population size is constant. To account for the changes in population size imposed by periodic bottlenecks, and also to take into account clonal interference, we recently extended (Campos and Wahl 2009) the theory proposed by Gerrish and Lenski (1998). We show that the rate at which beneficial mutations fix can be estimated as
where B(t) is the population birth rate at time t (which can, for example, be taken as , in units of individuals per generation) and 2rsbτe−rt is the probability of fixation of a single mutation that first arises in time t (t= 0, … , τ) during the growth phase (Wahl and Gerrish 2001; Wahl et al. 2002). Note that Ub is the beneficial mutation probability, the probability that each new individual carries a de novo beneficial mutation. (In the simulations, however, we use discrete nonoverlapping generations and take Ub to be a mutation rate, allowing multiple beneficial mutations to occur in a single reproductive event. Because Ub≤ 10−2 throughout our investigation, however, such events will be sufficiently rare that there is no quantitative difference between the two approaches.)
The variable I in equation (3) gives the number of interfering mutations, which corresponds to the expected number of superior mutations that a given mutation of value sb will encounter en route to fixation. Assuming that the occurrence of interfering mutations is a Poisson process, the term e−I is just the probability that no interfering mutations occur before fixation. The mean number of interfering mutations I is then given by , where the term 1/2 corresponds to the average fraction of the population that consists of wild-type individuals while the focal mutation is en route to fixation (see Campos and Wahl 2009 for a detailed discussion), Tfix is the mean number of generations a given beneficial mutation takes to fix, and λ(sb) is the rate of superior mutations, those with selective effect greater than sb, per generation, which is calculated as
(Note that neither λ(sb) nor Kb depend on t, the time during the growth phase at which the beneficial mutation first occurs.) In our previous work, we show that the time to fixation can be obtained through a diffusion approximation, and that in the limit of large population size
Including deleterious mutations: We now extend this work so as to incorporate the occurrence of deleterious mutations of arbitrary effect. In the absence of beneficial mutations, the population evolves toward the mutation–selection balance, in which the expected frequency of individuals carrying k deleterious mutations is given by a Poisson distribution with mean λ=Ud/sd (Haigh 1978). We define the expected frequencies in this distribution as fk=e−λλk/k!.
We will use kmin to denote the minimum number of deleterious mutations in any member of the population; we will also use the term “best-adapted class” to refer to the class of individuals carrying exactly kmin deleterious mutations. In the absence of back mutation, which we ignore, kmin can only increase over time, and it does so through two mechanisms. If the number of individuals in the best-adapted class, N0f0, is small, this class may be lost by random sampling from one generation to the next. This effect is the well-studied Muller's ratchet (Muller 1964). The rate at which Muller's ratchet operates has been the focus of sophisticated modeling efforts (Charlesworth and Charlesworth 1997; Gabriel et al. 1993; Rouzine et al. 2008c). In addition, if beneficial mutations first occur in individuals who carry more than kmin deleterious mutations, deleterious mutations may reach fixation through genetic hitchhiking (Rice 1987). We investigate this hitchhiking rate further in the results to follow.
In either case, kmin increases over time. Nonetheless, once the best-adapted class is lost, a “shifted Poisson” is reestablished (Gessler 1995; Gordo et al. 2002) with a characteristic timescale of 1/sd (Johnson and Barton 2002). In the shifted Poisson, the expected frequency of the class with a total of Nd=kmin+k deleterious mutations is zero for Nd < kmin, and is otherwise given by . Thus fk still gives the expected frequency of the class that carries k more deleterious mutations than the best-adapted class.
In Figure 1, we illustrate this shifted Poisson for a sample run of our simulation, in which both Muller's ratchet and hitchhiking occur. We note that the distribution of k is typically described by fk, although variations do occur as seen in the third panel of the figure. Supplemental animations of this process, in which the number of individuals with Nd deleterious and Nb beneficial mutations evolve in time, are available at: http://www.apmaths.uwo.ca/~lwahl/CI.html. In these animations, it is possible to watch the interplay between selective sweeps and Muller's ratchet, and to differentiate between the two. In particular, we note that when kmin increases through hitchhiking, the mutation–selection balance for the new beneficial mutation is simultaneously being established while the new mutation is en route to fixation.
In the sections that follow, we make the simplifying assumption that whenever kmin increases, the shifted Poisson distribution is reestablished before the next successful beneficial mutation occurs. This assumption clearly fails when Kb > sd. However when Kb is large the MM regime has typically been reached and our approach is no longer valid.
To compute the rate of substitution of favorable mutations, we use two different approaches: the first approach is based on the assumption that the probability of fixation can be approximated by using an “effective selection coefficient,” which describes the net advantage of the mutant carrying the beneficial mutation compared to the best-adapted class of individuals in the population. In the second approach, we derive a full multitype branching process solution. Both approaches hold irrespective of whether sb is greater or less than sd, however the case sd > sb is well-approximated by the first, simpler approach (Campos and Wahl 2009).
EFFECTIVE SELECTION COEFFICIENT APPROACH
Assume that the best-adapted class in the population has kmin deleterious mutations. We define the effective advantage sk of a beneficial mutation occurring in a genetic background with k+kmin deleterious mutations as
Note that this defines sk relative to the best-adapted class in the population, rather than relative to the average population fitness, allowing us to simplify the expression. As before, sb is drawn from probability distribution P(sb). In this case, we assume that the fixation probability can be approximated as Pfix(k) = 2rskτe−rt (Wahl and Gerrish 2001b). Following equation (3), we now write the expected rate of fixation Kb as
Here we define kmax as the maximum value of k, such that (1 +sb)(1 −sd)k > 1; for a specific value of sb, beneficial mutations that first occur in classes of individuals with up to kmax+kmin deleterious mutations have a fixation probability that is larger than the neutral case. Now following equation (4) we have
We note that another subtle approximation enters here. In particular, beneficial mutations with s < sb could in principle interfere, if they first occur in a class with fewer deleterious mutations than the focal beneficial mutation. Although it would be fairly straightforward to include this possibility by making λ a function of both sb and k, we have neglected it here.
The expected rate at which interfering mutations occur while a mutation of benefit effect sb is en route to fixation is then I= (1/2)λ(sb)Tfix, where the time of fixation Tfix is written as
Here we likewise assume that Tk has the same form as equation (5) with sb replaced by sk.
The case 1/α < sd is particularly interesting because in this situation the great majority of the beneficial mutations can only contribute to Kb if they first occur in an individual who carries no deleterious mutations, that is, kmax= 0. In this simple case, the rate of fixation of favorable mutations is
where f0= exp (−Ud/sd).
In Figure 2, we compare our simulation results to the theoretical predictions of the effective selection coefficient approach. In panel A, we find good agreement between theory and simulations, both for situations when a single deleterious mutation is enough to reduce sk below zero for most beneficial mutations sd= 0.1, α= 20 (eq. 11, solid line) or for situations when multiple deleterious mutations may be tolerated sd= 0.05, α= 10 (eq. 8, dashed line). However, quantitatively the theory starts to fail when the ratio Ud/sd becomes large (compare open circles and dashed line, panel B, for which Ud/sd= 5). Because this ratio gives the expected number of deleterious mutations per genome at the mutation–selection balance, many more classes contribute to the adaptation rate when Ud/sd is large, and in particular mutations that first occur in class k could be lost from class k due to the accumulation of more deleterious mutations, but might still fix in higher classes of higher k. This possibility is neglected in the effective s approach.
BRANCHING PROCESS APPROACH
For simplicity in the notation, in this section we treat the case when kmin= 0. The extension for kmin > 0 is straightforward, but unduly complicates the expressions. First, we define wk=er(1 +sb) (1 −sd)k. Assuming Poisson-distributed offspring, the offspring of a class k individual who also carries a beneficial mutation will thus have probability generating function (pgf) exp(wk(x− 1)). Each of these offspring will be in class k with probability , in class k+ 1 with probability , and so on, such that the pgf for the offspring of a class k individual, is
where is the column vector of dummy variables . In the absence of population bottlenecks, the vector of extinction probabilities for lineages that begin with a novel beneficial mutation in each class would then be the solution to .
To take population growth and bottlenecks into account, we simply compose the vector of pgfs, , with itself for each growth generation, compose the result with the bottleneck pgf (), and find the solution to:
where denotes functional composition, repeated τ times. The vector gives the extinction probability for lineages beginning at t= 0 during the growth phase. Extinction probabilities for lineages beginning at later times are then given by (Hubbarde et al. 2007). Although this approach does not, unfortunately, lend itself to closed-form approximation, it has the benefit of being numerically straightforward; simple iteration readily produces the solution , and follows directly.
The expected adaptation rate is then estimated as described above, except that we condition the fixation probability on k, the class in which the mutation first appears. Thus the expected rate of fixation is
where both Xt and I are functions of sb and k.
The rate at which interfering mutations occur is estimated by taking the product I(sb, k) = (1/2)λ(sb) Tk, where λ(sb) is the expected rate at which mutations with s > sb (and which will ultimately fix) occur per unit time
Note that Xt is a function of s and k.
From Figure 3 we see that the branching process approach typically underestimates Kb, but yields a better approximation for Ud/sd= 5 than the effective s approach (Fig. 2B, dashed line and open circles). In a more detailed exploration of parameter space, we found that the branching process consistently performed better when Ud/sd exceeded 1 or 2 (data not shown). This is presumably because the fixation probability for beneficial mutations that move between classes, while en route to fixation, is appropriately considered in the branching process approach.
Although we observe that deleterious mutations have a substantial effect on the rate of substitution, Kb, we also see (Figs. 2B and 3B), that keeping sd constant, this sensitivity to the deleterious mutation rate disappears at large Ub.
ADAPTATION EFFECTIVE POPULATION SIZE
To define an “adaptation effective population size” in the presence of clonal interference and deleterious mutations, we use the following primary equation, which estimates the rate at which mutations of selective advantage sb will occur and fix in the presence of clonal interference alone
where IC is the number of interfering mutations as previously defined. Following Gerrish and Lenski, we have for example
if we take and 2sb as the fixation probability.
We have previously demonstrated that for a population experiencing periodic population bottlenecks, the analogous substitution rate is given by
Including bottlenecks, clonal interference, and deleterious mutations, we have
Again, comparing with equation (17), and ignoring for the moment differences between IBCD and IC, this suggests the following adaptation effective population size:
where λ=ud/sd and (with the dependence on sb entering through the calculation of kmax). Finally, assuming that can be approximated by F as well, equation (21) simplifies to Ne≈N0r2τF(sb) (1 −ud/sb), which is only valid when ud < sb. Clearly an effective population size that depends on sb is unsatisfactory. Assuming that all beneficial mutations have the same effect, sb= 1/α, would yield
Under the (unrealistic) assumption that all beneficial mutations have the same effect, this adaptation effective population size should hold when αud < 1, when both sb and sd are small enough that a first-order Taylor series approximation of sk is valid, and when the least-fit class contributing to F is a small fraction of F. (Neglecting this last difference implies that Ne gives an underestimate of the true effective population size.)
Despite these weaknesses, Figure 4 demonstrates that replacing N in equation (17) with Ne from equation (22) gives a reasonable approximation for κBCD, underestimating the substitution rate but fitting simulated data surprisingly well over a wide range of Ub. We note that Ne has also been substituted in the expression for IC, and so the underestimate may also be in part because IBCD is smaller than IC when approximated in this way.
The range of the clonal interference regime: As described previously, delineating the parameter regime for clonal interference (CI) is not straightforward when a distribution of beneficial selective effects is assumed. Nonetheless, we can obtain a rough approximation for the value of Ub at which CI begins to take effect by using I, the mean number of interfering mutations, and Ne, the effective population size given in equation (22). To do this, we use the fact that CI is effective when the mean number of interfering mutations is greater than one. Of course, I depends on sb. However, here we will adopt an arbitrary criterion, assuming that CI becomes relevant when I for sb= 1/α is greater than one. Substituting our estimate of Ne for N in the analytical prediction of I (as done in Gerrish and Lenski (1998)), we find that for the set of parameters Figure 4A, CI starts at Ub≃ 5 × 10−6. As previously discussed, the MM regime begins when Kb > 1/Tfix. By replacing N with Ne and sb with 1/α in equation (5), we find that the the MM regime starts around Ub= 5 × 10−3. Although these are very rough estimates because our analytical prediction underestimates Kb on the boundary between the CI and MM regimes, they indicate that CI is effective over a broad range of Ub.
DEPENDENCE ON N0
One of the remarkable features of clonal interference is that at large population size N the rate of substitution approaches a constant (Wilke 2004), although the speed of adaptation, measured as the change of log fitness over time, and the mean effect of fixed mutations continues to grow with N. At this point, we investigate the dependence of Kb on N0, which is directly related to our definition of effective population size. Figure 5 displays the rate of substitution Kb as a function of Ub for several values of N0. For low beneficial mutation rates, increasing N0 increases the adaptation rate, because in this regime the competition among mutations in distinct lineages does not exist, and Kb depends linearly on the effective population size Ne. As one further increases Ub, the difference between the estimates of Kb for large and small N0 starts to decrease. For instance, one can observe that for intermediate values of Ub, the simulation results for N0= 8192 and N0= 16384 are nearly indistinguishable. Finally, at very large Ub, corresponding to the MM regime, simulation results indicate that Kb becomes independent of N0.
We also note that our analytical predictions (solid lines, sk approach) give a reasonable approximation to Kb over the range of the SSWM and clonal interference regimes. For example, as described earlier, for N0= 512, the clonal interference regime extends to about Ub= 10−3. However as the population size increases, MMs will occur at lower values of Ub. Figure 5 nicely demonstrates this effect: as N0 increases, the range in which the analytical prediction fits the data shifts increasingly to the left.
Here, we extend the previous simulation results and relax the assumption of constant selective effects, sd, for the deleterious mutations. Although most of the empirical studies regarding the distribution of selective effects have focused on adaptive mutations, it is also very important to characterize the distribution of fitness effects of detrimental mutations, especially whether one wishes to understand the role of purifying selection on variability in natural populations (Loewe and Charlesworth 2006; Loewe et al. 2006).
To investigate the impact of variation in deleterious effects, here we assume that the distribution of selective effects sd follows a gamma distribution. The gamma distribution is particularly convenient because of its two-parameter form, which enables us to create a broad range of curve shapes.
In Figure 6A, we compare the previous simulation results for fixed sd= 0.1 with two distinct simulations, both assuming variable sd. In the former, we take parameter values for the gamma distribution to be α= 1 (exponential distribution) and β= 10; and in the second we assume α= 5 and β= 50. In these two instances, the mean value of the distribution, α/β, is exactly 0.1. In Figure 6B, we present results for α= 1 and β= 50 and compare with fixed sd= 0.02. In all these simulations, we do not observe any noticeable difference between the simulations for fixed sd and variable sd.
FIXATION RATE OF DELETERIOUS MUTATIONS
Although the mean number of deleterious mutations in the population is kmin+Ud/sd, the mean number of deleterious mutations carried by mutations that ultimately fix, is biased to be smaller than this value. Thus, the rate of increase in kmin (the deleterious fixation rate) through hitchhiking is given by . Figure 7 illustrates this effect, plotting the fixation rate of deleterious mutations as a function of the fixation rate of beneficial mutations, Kb. The solid circles show results for a large population size, in which Muller's ratchet is not a concern, whereas the solid line shows the analytical prediction, , where has been estimated from simulation results. For comparison, the dashed line gives (Ud/sd)Kb. We also show the rate of change of kmin for a smaller population size, in which both Muller's ratchet and hitch-hiking occur. Interestingly, in this case the rate of increase of kmin is no longer linear in Kb. Estimating this rate presents an intriguing challenge for future work. Finally, we note that in the MM regime (high Kb), the solid line overestimates the rate dkmin/dt, which tends to saturate. This is evidence that the simultaneous fixation of two or more mutations does not affect the deleterious genetic background more than observed in the clonal interference regime.
We find that computing the effective selection coefficient, or using a full multitype branching process, can each provide reasonable predictions for the adaptation rate of asexual populations. Our results suggest that the effective s approach is more accurate, as long as the mean number of deleterious mutations, Ud/sd, is small. We have also derived an adaptation effective population size, which allows for a simple approximation of the adaptation rate, although it yields an underestimate of the true rate.
Throughout our analysis, we have assumed a distribution of beneficial selective effects, whereas all deleterious mutations have a single fixed effect. Simulations suggest, however, that assuming a distribution of deleterious effects has little impact on the adaptation rate, as long as the mean value of sd is unchanged.
Our formulation has been derived specifically for the clonal interference regime, although it remains valid at the lower mutation rates of the SSWM regime. In contrast, other approaches formulated for the MM regime remain valid throughout the clonal interference and SSWM regimes (Tsimring et al. 1996; Desai et al. 2007; Rouzine et al. 2008a,c). Unfortunately, solutions for the MM regime are necessarily more complex, and do not offer simple approximations for the fixation of beneficial mutations and deleterious mutations simultaneously, nor do they treat distributions of beneficial selective effects. Thus our formulation, and the adaptation effective population size in particular, may be of practical use in the experimental investigation of the adaptation rate.
Associate Editor: L. Meyers
This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco and program PRONEX/MCT-CNPq-FACEPE, and by the Natural Sciences and Engineering Research Council of Canada.