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Genetics, Vol. 174, 2071-2079, December 2006, Copyright © 2006
doi:10.1534/genetics.106.062406
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Zoology Department, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada
1 Corresponding author: Zoology Department, University of British Columbia, 6270 University Blvd., Vancouver, BC V6T 1Z4, Canada.
E-mail: rbarrett{at}zoology.ubc.ca
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ABSTRACT |
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 TOP ABSTRACT THE MODELNUMERICAL SIMULATIONSNUMERICAL RESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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Theory for determining the probability that selection will fix a new favorable mutation was first formulated 80 years ago by FISHER (1922, 1930) and HALDANE (1927), who focused on the fate of a single isolated mutation. In a recent flurry of articles, researchers have explored the distribution of fitness effects expected among the array of possible beneficial mutations that might arise within a population (GILLESPIE 1983, 1984, 1991; ORR 2002, 2003). This distribution can be seen as the starting point for progress toward a general theory of adaptive evolution. From this first distribution, we can determine the distribution of those mutations not lost by drift, referred to as "contending mutations" (GERRISH and LENSKI 1998; ROZEN et al. 2002). The distribution of contending mutations can subsequently be used to determine the distribution of mutations that outcompete other genotypes and fix in a population ("fixed mutations"), thus contributing to adaptation (ORR 1998; ROZEN et al. 2002).
In addition to theory, recent empirical studies have examined the steps from newly arisen mutations, to contending mutations, to fixed mutations. The underlying distribution of fitness effects of new beneficial mutations inferred from experiments is generally consistent with an exponential distribution (IMHOF and SCHLOTTERER 2001; ROZEN et al. 2002; SANJUAN et al. 2004; KASSEN and BATAILLON 2006), while the final distribution of fixed beneficial mutants appears roughly bell-shaped (ROZEN et al. 2002; ROKYTA et al. 2005; BARRETT et al. 2006). These empirical results must be interpreted cautiously, however, as there is typically little power to reject other distributions (e.g., more L-shaped or more bell-shaped distributions, see KASSEN and BATAILLON 2006).
An important caveat to the theoretical side of this work, however, is that it has assumed weak selection. R. A. Fisher (FISHER 1930) first justified this assumption using the analogy of movement from the outer surface of a sphere (representing phenotype space) to an optimum at the center; Fisher argued that mutations of small size have a 50% chance of bringing the population closer to the optimum, while larger mutations have a rapidly diminishing probability of being advantageous. This argument led to a "gradualist" view of adaptation, in which evolutionary change overwhelmingly proceeds through the selection of very slightly beneficial alleles (ORR 2005). Assumptions of weak selection common to all diffusion equations and many other theoretical approaches make it difficult to predict the dynamics of strongly beneficial mutations (MORJAN and RIESEBERG 2004). Simulations have demonstrated that although fixation times for strongly advantageous alleles are accurately predicted by diffusion, the probability of fixation is underestimated as the strength of selection increases (WHITLOCK 2003).
In recent empirical studies, researchers have observed mutations with very high selection coefficients (BULL et al. 2000; BARRETT et al. 2006), especially when organisms face novel environments. Strong selection has an important impact on the theory of adaptation. In particular, the fixation probability of mutations and the number of competing mutations will be highly dependent on the fitness effect of the mutation in question (ROZEN et al. 2002). When selection is assumed to be weak, new mutations remain at low frequency for a considerable period of time before reaching fixation (GERRISH and LENSKI 1998). This provides ample opportunity for beneficial mutations to be lost and for competing mutations to arise (GERRISH 2001). In contrast, when selection is strong, the probability of fixation approaches its maximum value of one and the time to fixation is relatively short, reducing the number of competing mutations and the importance of clonal interference. Yet previous predictions cannot be applied to the case of strong selection, because of the pervasive theoretical assumption that selection is weak.
Here, we derive population genetic theory to describe the impact of drift and clonal interference on the fitness distribution of fixed beneficial alleles without the assumption of weak selection. We derive these distributions for a wide range of selection coefficients and test our analytical theory against numerical simulations.
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THE MODEL |
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TOPABSTRACTTHE MODELNUMERICAL SIMULATIONSNUMERICAL RESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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The distribution of beneficial mutations:
It is generally assumed that the wild type has very high fitness and almost all mutations are deleterious (GILLESPIE 1983, 1984; ORR 1998). It follows that beneficial mutations will lie in the extreme right tail of a distribution of all mutant fitness effects. This inference justifies the application of extreme value theory (GUMBEL 1958) to describe the distribution of beneficial mutant effects. Extreme value theory suggests that the distribution of mutant effects, restricted to beneficial mutations, will be nearly exponential (GILLESPIE 1983, 1984). This requires, however, that only a tiny minority of mutations are beneficial. In very harsh or novel environments, mutations that were previously deleterious may become beneficial, thus increasing the size of the beneficial mutant class. In these situations, extreme value theory may not hold and an exponential distribution might not be an adequate description of the selective effects of new mutants.
While empirical and theoretical studies indicate that the exponential distribution is a plausible distribution describing the fitness effects of new beneficial mutations, distributions with other shapes cannot be rejected and might be more appropriate under certain circumstances. Thus, to allow greater flexibility, we assume that the selection coefficients of new beneficial mutants, s, follow a gamma distribution with mean selection coefficient, 
, and coefficient of variation, cv,
 | (1) |
[a] is Euler's gamma function. The shape of the gamma distribution varies from L-shaped (high cv) to bell-shaped (low cv), allowing a broader range of distributions to be described. The exponential distribution represents a specific case of the gamma where the coefficient of variation equals one.
The distribution of contending beneficial mutations:
We begin by deriving the distribution of fitness effects among those contending mutations that survive stochastic loss while rare. HALDANE (1927) used a branching process to show that P, the probability of fixation, satisfies 1 – P = e–(1+s)P in populations of constant size when the number of offspring per parent is Poisson distributed. Using a diffusion approximation, KIMURA (1957, 1964, 1983) extended this theory for populations of finite size, N, showing that 
. For weak selection (1/N << s << 1), both of these equations yield the same approximate fixation probability: P 
2s. When strong selection is possible, however, a more accurate approximation for both equations is: P = 1 – e–2s, obtained by letting N get large in the diffusion result (Figure 1). These results assume that a mutant allele is either lost or fixed before other mutations arise. When the mutation rate is sufficiently high, however, a mutation may survive loss while rare, but eventually be outcompeted by mutations arising in the future. In this case, P describes the probability of surviving stochastic loss while rare, not the ultimate fixation probability, which depends on the nature of future mutations.
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1 – e–2s in populations of large size, the distribution of selection coefficients among contending mutations becomes
 | (2) |
:
 | (3) |
 | (4) |
The distribution of fixed beneficial mutations:
ROZEN et al. (2002) defined the expected number of contending mutations arising within other genetic backgrounds before the fixation of a focal mutation as
 | (5) |
reflects the fact that, by symmetry, half of the population will not carry the focal mutation when averaged over the period of time during which the focal mutation rises from a single copy to fixation; see supplemental Figure 1 at http://www.genetics.org/supplemental/). ROZEN et al. (2002) then calculated 
under the assumption of weak selection. To relax this assumption, we use the general solution for the deterministic haploid model, q(t)/p(t) = (1+s)tq(0)/p(0), where p(t) and q(t) are the frequencies of nonmutant and mutant individuals, to solve for the time taken for an allele initially at frequency 1/N to reach a frequency of 
This gives 
, which can be used along with 
in (5) to estimate the number of contending mutations:
 | (6) |
When there are n contending mutations and one focal mutation, each with a selection coefficient drawn from a cdf given by G(s), the cumulative density function of the highest of the selection coefficients is G(s)n+1 (RICE 1988). Assuming that the number of contending mutations that appear during the spread of a focal mutation follows a Poisson distribution, with mean 
(s), the cdf for the selection coefficient of the most advantageous of the contending and focal mutations is
 | (7) |
is the average probability of fixation (Equation 3) and
 | (8) |
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NUMERICAL SIMULATIONS |
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TOPABSTRACTTHE MODELNUMERICAL SIMULATIONSNUMERICAL RESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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and coefficient of variation cv and assigned a unique identifier. Multiple mutations had independent effects on fitness (no epistasis on a multiplicative scale). Offspring were then sampled with replacement according to a multinomial distribution from the parental distribution of genotypes, weighted by the fitness of these genotypes. A fixation event was defined as the first point in time when all individuals in the population shared a common mutation (with the same identifier). Similar results were obtained when we recorded data for the fifth mutation rather than the first mutation to fix (data not shown). At this point, the process was stopped and the selection coefficient of the fixed mutant was recorded. This selection coefficient was defined as the fitness effect of the fixed mutant when placed in the ancestral background; the average fitness within the population at the time of fixation was also recorded.
To evaluate the robustness of the analytical results, we ran simulations with every combination of the following average selection coefficients ( = 0.01, 0.1, 1, 2, and 10), beneficial mutation rates (µ = 10–5, 10–7, and 10–9), population sizes (N = 105, 106, and 107), and coefficients of variation (cv = 0.5, 1, and 2). Simulations were carried out in Mathematica (WOLFRAM RESEARCH 2005; available upon request).
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NUMERICAL RESULTS |
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TOPABSTRACTTHE MODELNUMERICAL SIMULATIONSNUMERICAL RESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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, and cv (inset boxes). The shape of the contending and fixed distributions is also influenced by the shape of the distribution of underlying beneficial mutations (supplemental Figure 2 at http://www.genetics.org/supplemental/). Increasing the coefficient of variation of beneficial mutations results in more contending mutations of large and small effect, increasing the variation observed among fixed mutations.
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, and the distribution of fixed mutations, h(s). |
DISCUSSION |
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TOPABSTRACTTHE MODELNUMERICAL SIMULATIONSNUMERICAL RESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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The distribution of fitness effects among fixed beneficial mutations is generally derived from the distribution of fitness effects among all possible beneficial mutations, about which little is known. Several theoretical studies have suggested that new beneficial mutations should be exponentially distributed (MUKAI et al. 1972; ORR 1998; ROZEN et al. 2002; WILKE 2004), on the basis of the fact that beneficial mutations represent the tail of the distribution of potential mutant effects (GILLESPIE 1983, 1984). In a novel environment, however, more mutations are likely to be beneficial and the applicability of such extreme value theory is uncertain. We have thus employed a gamma distribution to describe the fitness effects of possible beneficial mutations. Because the gamma distribution has two parameters (described by the mean selection coefficient, , and the coefficient of variation, cv), we can explore a broader range of possible distributions of mutational effects. We find that the shape parameter of the gamma distribution among newly arising mutations influences the distribution of mutations that survive stochastic loss while rare (contending mutations) and the distribution of mutations that survive clonal interference to become fixed (fixed mutations), especially when selection is strong (supplemental Figure 2 at http://www.genetics.org/supplemental/). This result appears to contradict a recent study, which reported that the distribution of mutational sizes for fixed mutations is virtually independent of the underlying distribution of beneficial mutations (ORR 1998; HEGRENESS et al. 2006). The simulations run by HEGRENESS et al. (2006) cover only a range of parameters, within which clonal interference is severe (the population size was set to 
and the mutation rate was 10– 5). Indeed, using their combination of parameter values in Equation 7 indicates that the shape of the distribution of fixed beneficial mutations is nearly independent of the shape of the distribution of newly arising beneficial mutations. Furthermore, using their parameters, most fixed mutations have similar selection coefficients, as pointed out by HEGRENESS et al. (2006), unlike the fairly broad distributions observed in Figure 3.
Beneficial mutations that survive stochastic loss while rare tend, on average, to have a larger fitness benefit, and their distribution tends to have a lower coefficient of variation (more bell-shaped), because very weakly selected alleles are unlikely to fix (KIMURA 1983; GERRISH and LENSKI 1998; ORR 2000; OTTO and JONES 2000; ROZEN et al. 2002; WILKE 2004). While this is generally true, the effect is less pronounced when selection is strong. That is, the distribution of fixed beneficial mutations is more similar to the distribution of newly arising mutations (Figure 3). Consequently, for empirical data involving high selection coefficients, using theory that assumes weak selection will tend to underestimate the mean selection coefficient among newly arising beneficial mutations.
One of the major impediments to theoretical studies of the distribution of fitness effects of fixed beneficial mutations has been a lack of knowledge of realistic parameter values. This is largely because the low frequency of fixed beneficial mutations has prevented empirical work with statistical power. In recent years, however, the use of microbial microcosms has provided a way to increase the number of beneficial mutations likely to arise and fix during an experiment. Three experiments have characterized the distribution of fixed beneficial mutant effects (ROZEN et al. 2002; ROKYTA et al. 2005; BARRETT et al. 2006). All used roughly the same experimental protocol: a number of replicate bacterial or viral lines were introduced into a novel environment and evolution proceeded through the substitution of novel beneficial mutations. By comparing the fitness of an evolved genotype sampled from around the time a mutation fixed within the evolved population to the fitness of the ancestral genotype, these studies claimed to measure the fitness advantage conferred by the single beneficial mutation carried by each evolved genotype. Our simulations indicate, however, that the selection coefficient estimated from the average fitness of individuals at the time of fixation is a very poor measure of the fitness effect of the actual mutation that has just fixed whenever Nµ > 1 (Figure 4). Whenever multiple mutations arise during the spread of a focal mutation, so that clonal interference occurs, individuals are likely to carry multiple mutations by the time that the focal mutation has fixed, and therefore selection coefficients measured will overestimate the true effects of a single mutation. Consequently, experiments aimed at estimating the distribution of beneficial selective effects should avoid large population size to keep Nµ < 1.
The studies by ROZEN et al. (2002) and BARRETT et al. (2006) serve as a good comparison of how the distribution of fitness effects of fixed beneficial mutations shifts with increasing selection. Both experiments were conducted with similar organisms and transfer protocols and therefore share fairly comparable parameter values, except for the average selection coefficient, which differed by an order of magnitude. This difference is perhaps unsurprising as the ancestral strain in the ROZEN et al. (2002) experiment was fairly well adapted to consuming the sole carbon source (Escherichia coli with glucose), whereas the ancestral strain in the BARRETT et al. (2006) experiment initially had very poor growth (Pseudomonas fluorescens with serine). Despite the difference in average selection coefficients, both studies reported bell-shaped distributions for fixed beneficial mutations. Both sets of authors suggest that this shape is the result of drift and clonal interference transforming an exponential distribution of beneficial mutations. However, since Neµ < 1, the bell-shaped distributions are unlikely to be strongly influenced by clonal interference. Indeed, the bell-shaped distributions observed in these studies can be accounted for entirely by the stochastic loss while rare of mutations drawn from an exponential distribution (Equation 4), without considering competing mutations.
In conclusion, allowing for strong selection has altered our theoretical understanding of the distribution of fitness effects in the following ways. By correctly accounting for the fact that the fixation probability cannot rise above one, the distribution of fixed beneficial mutations more closely matches the distribution of newly arising beneficial mutations when selection is strong (Figure 3). Although the distribution of surviving mutations is always more bell-shaped, the difference from the distribution of newly arising beneficial mutations is largely confined to regions where selection is weak. Importantly, our results (Figure 4) also demonstrate that data on the selection coefficients of fixed mutations must be treated with caution whenever clonal interference is present, as multiple mutations are likely to be segregating at the time of fixation, causing selection coefficients to be greatly exaggerated.
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APPENDIX |
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TOPABSTRACTTHE MODELNUMERICAL SIMULATIONSNUMERICAL RESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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(EWENS 1967; KIMURA and OHTA 1974; OTTO and WHITLOCK 1997). Here, the arithmetic average population size is 
, and the "effective" population size is Ne, whose calculation depends on the nature of the population fluctuations (OTTO and WHITLOCK 1997; WAHL et al. 2002). Unfortunately, we lack an analytical expression for the fixation probability when selection is strong and population size varies. We conjecture that an adequate approximation for the average fixation probability under strong selection is given by 
, which is nearly when selection is weak but has the advantage of remaining <1 when selection is strong. This approximation is equivalent to the one used when the population size is constant; i.e., 
(see Figure 1). This functional form is also suggested by diffusion analysis in populations of large effective size (KIMURA 1957, 1964), which assumes weak selection. Simulations confirm that 
provides a satisfactory approximation for the fixation probability over a range of parameter values in populations undergoing repeated bottlenecks (within a factor of two; supplemental Figure 3 at http://www.genetics.org/supplemental/).
We next consider the time to fixation of a beneficial mutation. If the mutation arises when the population size is 
and fixes when the population size is 
, a deterministic model of selection can again be used to predict that
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 | (A1) |
is the geometric mean population size over time. In Equation A1, 
represents the period of the population size cycle if population size changes cyclically. If not, Equation A1 is evaluated by taking the limit as goes to infinity. Simulations indicate that 
provides a satisfactory approximation for the average time to fixation over a range of parameter values in populations undergoing repeated bottlenecks (within a factor of two; supplemental Figure 4 at http://www.genetics.org/supplemental/).
To account for clonal interference, we should determine the expected number of mutations that compete for fixation when the focal mutation appears at time t (see Equation 5) and then average over all possible times at which the focal mutation could arise. To do so exactly requires a precise description of the manner in which the population size fluctuates. As a first-order approximation, we estimate the number of competing mutations using
 | (A2) |
Using Equation A2 to rederive Equation 7, the cdf among fixed beneficial mutations becomes
 | (A3) |
is again given by Equation 8 and the average probability of fixation across the distribution of new mutations is now
 | (A4) |
.
We assessed the accuracy of Equation A3 against simulations of a population whose size cycles from N0 to 27N0 via seven doubling events followed by a 1/27 serial dilution. In these simulations, the growth of the population was assumed to be deterministic (no sampling except during the dilution or "bottleneck" generation), and births occurred at a rate proportional to the fitness of an individual. Under this scenario, the size of the bottleneck, N0, and the period of the cycle, , determine 
(WAHL et al. 2002), 
, and 
for use in Equation A3. Every combination of the following parameters was explored: selection coefficients ( = 0.01, 0.1, 1, 2, and 10), beneficial mutation rate (µ = 10–7 and 10–9), and initial population size (N0 = 105, 106, and 107), assuming that the fitness effects of new mutants were exponential (cv = 1).
Figure A1 indicates that Equation A3 accurately predicts the distribution of fixed selective effects across this range of parameters. Interestingly, Equation 2 of ROZEN et al. (2002) provides a more accurate prediction of the distribution of fixed beneficial mutations with a fluctuating population size (with 
in place of N) than with a constant population size (Figure 2). The improved performance of their method is due to the fact that the fixation probability used, , remains reasonably accurate even when selection is strong (
) because of the reduction in effective population size caused by the fluctuations (
).
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ACKNOWLEDGEMENTS |
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TOPABSTRACTTHE MODELNUMERICAL SIMULATIONSNUMERICAL RESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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LITERATURE CITED |
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TOPABSTRACTTHE MODELNUMERICAL SIMULATIONSNUMERICAL RESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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, relative to the selection coefficient of that mutation alone, 
. The horizontal axis gives the input rate of new mutations, Nµ. Results are based on 200 replicate simulations.