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Proc Natl Acad Sci U S A. Apr 10, 2007; 104(15): 6266–6271.
Published online Apr 3, 2007. doi:  10.1073/pnas.0607280104
PMCID: PMC1851075

Complete genetic linkage can subvert natural selection


The intricate adjustment of organisms to their environment demonstrates the effectiveness of natural selection. But Darwin himself recognized that certain biological features could limit this effectiveness, features that generally reduce the efficiency of natural selection or yield suboptimal adaptation. Genetic linkage is known to be one such feature, and here we show theoretically that it can introduce a more sinister flaw: when there is complete linkage between loci affecting fitness and loci affecting mutation rate, positive natural selection and recurrent mutation can drive mutation rates in an adapting population to intolerable levels. We discuss potential implications of this finding for the early establishment of recombination, the evolutionary fate of asexual populations, and immunological clearance of clonal pathogens.

Genomic mutation rates are governed by genes that are themselves subject to mutation (1). Such mutations often increase (“mutators”) and rarely decrease (“antimutators”) genomic mutation rate (2), thereby altering the rate at which mutations affecting fitness arise. In the absence of genetic exchange, mutators and antimutators remain linked to the fitness mutations they produce. In a population, therefore, mutators and antimutators can change substantially in frequency as a consequence of their effect on the production of fitness mutations (3). The ultimate consequence of this process of indirect selection on the mutation rate is the subject of our investigation.

In general, mutations affecting fitness are much more likely to be deleterious than beneficial (4), but the rare beneficial mutations that do occur are much more likely to achieve long-term evolutionary success than deleterious mutations (5, 6). Although mutators will be selected against most of the time because of their association with increased numbers of deleterious mutations, mutators also produce beneficial mutations at a higher rate, and association with successful beneficial mutations can cause mutators to rise in frequency, thereby increasing the average mutation rate of a population. Such “mutator hitchhiking” certainly occurs in real populations (3, 5, 7). On the other hand, a rise in frequency of antimutators is likely to be slower than the hitchhiking of mutators (7, 8). Antimutators are certainly expected to have a substantial long-term selective advantage over mutators because of their reduced association with deleterious mutations. However, only direct fitness effects or decreased production of lethal mutations can give antimutators an immediate selective advantage over mutators because antimutators will tend to arise on genetic backgrounds that contain as many nonlethal deleterious mutations as the rest of the population (7, 8). Over time, the reduced association of antimutators with nonlethal deleterious mutations will give them an increased selective advantage over mutators, but this difference will be slow to evolve (5, 9). These considerations have led to the conclusion that the mutation rate should have an evolutionary tendency to increase in an adapting asexual population (5).

We ask, where will the tendency toward increased mutation rate ultimately take an asexual population? Previous workers have suggested that mutation rate increase should be contained by either the cost of more abundant deleterious mutations or intensifying competition between beneficial mutations (5, 10). Here, we present analytical models and simulations that support a different conclusion: under broadly defined conditions, the mutation rate can increase without bound in an adapting asexual population until it reaches an intolerable level that would presumably cause extinction. [Henceforth, all claims of extinction should be understood as being the inferred consequence of high mutation rates in conjunction with a rapid drop in fitness. See supporting information (SI) Text and refs. 11 and 12.] There is almost certainly no physiological barrier to such an effect in most organisms: the genomic mutation rate in organisms from viruses to eukaryotes is a quantitative trait affected by many mutations whose effects can readily cumulate to intolerable levels of error (1, 13). In what follows, we show that there need not be a selective barrier to this process either: because the full fitness effect of increased deleterious mutation takes some time to accumulate after a higher mutation rate has evolved, it is theoretically possible for a population to evolve a critically high mutation rate and subsequently go extinct.


Analytical Models.

We consider two analytical models of evolution in asexual populations in which mutations affect both fitness and mutation rate:

equation image

where u = u(x, y, t) is the density of individuals whose log fitness is x and whose log genomic mutation rate is y at time t, measured in generations; [x with macron] denotes mean log fitness ([x with macron] = ∫∫xu(x, y, t)dxdy) at time t. The mutation operator M models mutation to and from other genotypes in the population. Mutations can affect either fitness, represented by a change in x, or mutation rate, represented by a change in y. Genomic mutation rate of an individual of genotype (x, y) is ey. To clarify the mechanics of M, we give the following example. A deleterious mutation may decrease fitness by an amount Δx, where the parent's fitness is x + Δx and the offspring's fitness is x. The probability density for Δx is gDx), and the mutational “flux” from fitness class x + Δx to fitness class x is eyfDu(x + Δx, y, t)gDx), where fD is the fraction of mutations that are deleterious. The total mutation influx of deleterious mutations into fitness class x is

equation image

The total outflux of deleterious mutations from fitness class x is OD = eyfDu(x, y, t), and the net change in u(x, y, t) due to deleterious mutations is therefore

equation image

Similar logic gives terms for beneficial, mutator, and antimutator mutations, and all of these terms are then summed to construct mutation operator M, which is then defined by

equation image

where fB is the fraction of mutations that increase fitness (beneficial), fD is the fraction of mutations that decrease fitness (deleterious), fM is the fraction of mutations that increase mutation rate (mutator), fA is the fraction of mutations that decrease mutation rate (antimutator), gB is the distribution of effects of beneficial mutations, gD is the distribution of effects of deleterious mutations, gM is the distribution of effects of mutator mutations, and gA is the distribution of effects of antimutator mutations.

If the distributions of mutational effects, g, have means m and variances σ2 then second-order Taylor expansion of M gives

equation image


equation image

The two models differ only in their assumptions about when mutation occurs and numerical solution shows they give qualitatively equivalent results. Model 1 most closely mimics organisms whose offspring may suffer mutations independently of one another (e.g., binary fission of bacteria, many viruses, and eukaryotes in general). Model 2 most closely mimics organisms whose offspring are produced en masse from a template genotype and for which mutation is most likely to occur in the replication event that creates the template (e.g., retroviruses).

The two models lend themselves naturally to two qualitatively different but complementary analyses giving the eventual dynamics and the ultimate state of the system (SI Text). First, we analyzed Model 1, to which we sought a dynamic limiting solution of the form

equation image

[Previous work employed a similar approach to model the evolution of fitness only, as u(x, t) → û(xct); see refs. 14 and 15.] Figuratively, this solution assumes that the population distribution asymptotically converges to a distribution that appears unchanging to an observer who is moving along the fitness axis at velocity c. If such a solution exists and supports c > 0, this would confirm the established notion that mutation rate converges to some optimal or stable distribution that balances adaptability and adaptedness (10, 16, 17) while fitness steadily increases. However, our analysis concludes that such a balance is not achieved: the proposed solution is only consistent with strongly negative c and a very flat distribution, suggesting that the population ultimately decreases rapidly in fitness and becomes genetically very heterogeneous. Next, we analyzed Model 2, to which we sought a static limiting solution of the form

equation image

determined by setting [partial differential]u/[partial differential]t = 0. Our analysis shows that ũ(x, y) decreases monotonically in x and increases monotonically in y, supporting the conclusion that log fitness ultimately decreases without bound while mutation rate continues to increase. Taken together, analyses of Models 1 and 2 show that a population will eventually experience an abrupt transition to its final state of very high mutation rate and zero fitness (Fig. 1A), a recipe for certain extinction (SI Text and refs. 11 and 12). The conclusions stated for both of the above analyses are true under the following two conditions: (i) fMmM > fAmA: roughly, mutator mutations must be more common than antimutator mutations (sufficient but not necessary in both analyses), and (ii) fDmD > fBmB: roughly, deleterious mutations must be more common than beneficial mutations (sufficient but not necessary in the first analysis; sufficient and necessary in the second).

Fig. 1.
Representative plots of observed fitness (blue lines) and mutation rate (red lines) dynamics. (A) A numerical solution of Model 1 (infinite population) with parameters μ = 0.003, fB = 10−6, fD = 0.1, fM = 0.001, fA = 0.0001, and mB = ...

Individual-Based Simulations.

Our analytical approach assumed infinite population size. To examine the effects of finite population size, we conducted computer simulations of asexual populations in which both mutation rate and fitness were subjected to mutational change (Materials and Methods). These simulations, for which code and executable files are available in SI Text, kept track of every individual and every replication event in the population. Numbers of each kind of mutation acquired by each replication event were Poisson distributed, as were total numbers of offspring. The effects of each kind of mutation were governed by continuous distributions. For the figures exponential distributions were chosen. However, analytical theory and simulations show that the qualitative outcome is robust to the choice of these distributions. At the outset, simulated populations were homogeneous for fitness and mutation rate.

When beneficial mutations were absent from our simulations, we observed that average fitness declined gradually and monotonically as existing models of Muller's ratchet predict (18, 19), and average mutation rate, if anything, decreased as predicted by previous models of mutation rate evolution in nonadapting populations (20). When beneficial mutations were included, however, we observed that both average fitness and mutation rate increased over time. Eventually, mutation rate increase began a marked acceleration, and fitness declined suddenly and precipitously (Fig. 1B). In other simulation runs, we varied population sizes, distributions of mutational effects, relative rates of mutation to beneficial and deleterious mutations, and relative rates of mutation to antimutator and mutator alleles systematically over broad ranges that included biologically realistic values (Materials and Methods). Fig. 2 reports extinction times for these simulation runs.

Fig. 2.
Sensitivity of extinction times (all vertical axes) to various parameters. Red corresponds to Wright–Fisher simulations (SI Text) of populations of size N = 1,000. Green corresponds to regular simulations with population size N = 10,000. Default ...

Integral Forms Provide Link to Classical Theory and Help Decipher the Extinction Result.

Our observations are consistent with previous theoretical and experimental work demonstrating that a high mutation rate can cause extinction (21, 22). Our findings depart from previous work, however, by showing that such high mutation rates can be the catastrophic result of unfettered natural selection. The underlying processes that give rise to such a “mutation rate catastrophe” are elucidated by two representations of Model 1 obtained by multiplying by x and ey, respectively, and integrating over x and y

equation image

where μ = ey, genomic mutation rate. These expressions are modifications of Fisher's fundamental theorem (4) and the Price equation (23), respectively, and these infinite-population results, quite surprisingly, show strong agreement with the individual-based simulations of finite populations (Figs. 1B and and3,3, and SI Text).

Fig. 3.
Comparing analytical theory with individual-based simulations. Data were taken from a simulation run with a population size of 10,000. (A) Predicted values for [partial differential][mu]/[partial differential]t were obtained by computing cov(μ, x) + fMmM ...

Linkage between mutators and rare beneficial mutations (hitchhiking) causes positive association between fitness and mutation rate [cov(μ, x) > 0]. The resulting increase in mean mutation rate, [partial differential][mu]/[partial differential]t > 0, is only exacerbated by mutation pressure (fMmMμ2), and mutation rate continues to increase despite its detrimental effect on fitness increase (−fDmD[mu]). Fig. 4A shows how hitchhiking and mutation pressure conspire to relentlessly elevate mutation rate. Eventually, fMmMμ2 and −fDmD[mu] become the dominant terms, triggering the population's transition: fMmM[mu]2 overwhelms the now negative association between fitness and mutation rate [cov(μ, x) < 0], making [partial differential][mu]/[partial differential]t strongly positive and [partial differential][x with macron]/[partial differential]t strongly negative. In finite populations, the mutation rate marches slowly upward through successive hitchhiking events, eventually causing the transition (Fig. 1B). By contrast, in infinite populations, the mean mutation rate stays relatively constant while rare, high mutation-rate variants accumulate; eventually, one giant hitchhiking-like event occurs, causing the very sudden transition (Fig. 1A). Animated contour plots of numerical solutions (SI Movies 1 and 2) further elucidate the underlying processes leading to this sudden transition: there, one can see the slow accumulation of mutators and the hitchhiking effect, shown by the fact that mutators appear to lead the fitness increase [cov(μ, x) > 0].

Fig. 4.
The mechanics of extinction. (A) Mutation rate partitioned into its two theoretical components: the hitchhiking component (HC, blue line) and the mutation component (MC, green line). The predicted mutation rate (thin black line) is the sum of these two ...

Time to Extinction.

In the infinite population limit, this transition is manifest as a mutation rate singularity that occurs at time

equation image

denoted the “extinction time,” where h = cov(μ, x), the average covariance before the transition (or “hitchhiking index”), and μ0 is the initial genomic mutation rate (SI Text). The zero covariance limit gives a conservative upper bound: tE < (μ0fMmM)−1.


How Genetic Linkage Can Subvert Natural Selection.

Our theoretical findings indicate that mutator hitchhiking can set in motion a self-reinforcing loss of replication fidelity, but the question of how a process as robust as natural selection could allow this to happen remains. The key fact is that natural selection, although eminently robust, is a short-sighted process that favors traits with immediate fitness benefits. The fitness cost of mutator hitchhiking is generally not anticipated because of the slow accumulation of deleterious load. When a mutator hitchhikes with a new beneficial mutation, a simple model shows that the increased deleterious load due to the mutator is in fact suppressed during the spread of the beneficial mutation. Indeed, the full fitness cost of the mutator is only realized well after the beneficial mutation has stopped spreading (SI Text). A mutator may therefore enjoy the immediate benefit of producing a new beneficial mutation without anticipating the eventual increase in deleterious load. Because of this delay in the accumulation of deleterious load, natural selection can drive mutation rate up to the point of no return, where fMmMμ2 becomes the dominant term (Fig. 4A); even if the increase in deleterious load is lethal, it is not anticipated (Fig. 4B). At the population level, this failure to anticipate the establishment of a lethal deleterious load is partly due to the sharpness of the threshold separating lethal from viable mutation rates (22, 24), such that there is no slow fitness decrease to “warn” of impending extinction.

Model Assumptions.

The mutation rate catastrophe that we have modeled is a result of very general evolutionary factors acting in conjunction: linkage, adaptation, and the existence of a maximum tolerable mutation rate. The simulation and analytical approaches that we have employed are based on some additional assumptions: First, although a significant fraction of deleterious mutations may be immediately lethal without changing our results qualitatively (SI Text and ref. 25), we assumed that not all deleterious mutations are immediately lethal; this is well supported by numerous studies of deleterious mutations in a variety of taxa (26). Second, we assumed that mutation rate can increase incrementally to intolerable levels, by accumulation of mutations in replication and/or repair genes that are not directly lethal. This second assumption is reasonable given that genomic fidelity is affected by enzymatic processes and pathways, both essential and nonessential, that could be subtly altered in many different ways. Indeed, recent experimental studies in Escherichia coli have shown that mutations in the proofreading subunit of the major replicative polymerase alone can be sufficient to elevate the genomic mutation to an intolerable level without directly affecting the capacity of the cell to replicate and survive (13, 27). Moreover, fundamental biochemical studies of replication in RNA viruses, which already have genomic rates close to intolerable levels, suggest obvious ways in which mutations could cause further decreases in replication fidelity without directly impairing the capacity of the virus to reproduce and survive (ref. 28 and C. Cameron, personal communication). Finally, we assume that antimutator mutations do not become more common than mutator mutations in the time frame required for extinction of the population. In a finite genome, as mutation rate increases: (i) the effective rate of mutator mutation should decrease, as ways to increase the mutation rate without directly killing the organism are depleted, and (ii) the effective rate of antimutator mutation should increase, as new ways to decrease the mutation rate (through reversion of mutators and compensatory mutation) should appear. In our initial exploration (Finite Genome Effects in SI Text), we find that the qualitative outcome is indeed sensitive to how robust the replication/repair apparatuses are (the inverse of how many mutations it takes to knock them out). Under the conservative assumption that replication/repair apparatuses are slightly less robust than the proteins subtilisin and TEM1 β-lactamase, whose robustness has been measured directly (29), we show that the mutation rate catastrophe still occurs (SI Text).

Empirical Support and Experimental Directions.

The two basic components necessary for the mutation-rate catastrophe to occur have been observed in natural and laboratory populations: (i) mutator hitchhiking has been directly observed in laboratory populations (8, 30) and inferred from natural isolates (31), and (ii) mutation rates have been elevated to levels that are not directly lethal but cause population extinction through increased production of deleterious mutations (13, 21, 22, 32). These observations indicate that at least some of the conditions required for the mutation rate catastrophe are present in natural populations. However, empirical corroboration of our theory must come from direct observation of the evolution of intolerable mutation rates in an adapting population and subsequent extinction. A step in this direction would be the observation of successive mutator hitchhiking events. Our previous experimental work (30) has documented single instances of mutator hitchhiking in experimental populations of E. coli during 10,000 generations of experimental evolution, but we have not observed any successive hitchhiking events in these populations after 25,000 generations of further evolution (unpublished observations). However, because these E. coli experiments are conducted in a constant environment, the rate and magnitude of effect of beneficial mutations decline greatly after the first few thousand generations, making successive mutator hitchhiking events unlikely. Our models predict that successive mutator hitchhiking events should be much more likely in environments that change, however slowly or erratically, such that adaptation is sustained, on average, in the long term. An intuitive segue to our previous experimental work (30), therefore, would be to introduce those mutator strains into a novel environment, let them adapt for many hundreds of generations, and observe whether they acquire a second mutator mutation.

Biomedical Implications.

The use of mutagenic agents to induce error catastrophe in viral populations has received attention as a possible therapeutic innovation (21, 32). Our results suggest the interesting and related possibility that the adaptive immune response itself could drive a purely clonal pathogen to mutation rate catastrophe and extinction within the host. During a humoral immune response, B cells mutate their immunoglobulin variable region genes at a high rate, a phenomenon called somatic hypermutation (33, 34). B cells that recognize antigen with high affinity tend to be selected, whereas those that recognize antigen with lower affinity tend to undergo apoptosis. If a pathogen evolves quickly, it can evade the immune response. However, because the immune response can also evolve, this leads to further evolution of the pathogen. Consequently, the immune response and pathogen can potentially accelerate each other's rate of evolution. This self-reinforcing state of accelerated evolution is an example of the “red queen” phenomenon (35), and our findings suggest that it could ultimately drive a truly clonal pathogen extinct. Here, we specifically model the red-queen dynamics of virus evolution and an adaptive immune response

equation image

where [var phi] = [var phi](x, t) is the density, at time t, of B cells and antibody specific for pathogen variants whose log fitness is x; u = u(x, y, t) is the density of pathogen whose log fitness is x and whose log mutation rate is y at time t; a is an immune activation coefficient; d is the effective rate of loss of antigen-specific B cells and antibodies; μ is the rate of somatic hypermutation under affinity maturation; σ2 is the variance in effects of hypermutation (symmetric mutation is assumed); and β characterizes the rate at which the immune response eliminates the pathogen. Fig. 5 shows that realistic conditions for in vivo populations of an RNA virus do not preclude the error-catastrophe outcome, suggesting that a truly asexual RNA virus could potentially be driven extinct by this mechanism. (In this highly simplified model, we eliminate the dimensions that describe antigenic character by mapping all of its features onto fitness. This simplifying assumption is eliminated in ongoing work, but the more realistic models show similar dynamics and also culminate in error catastrophe). This mechanism could, in theory, help to explain the spontaneous clearance of some viral infections and suggests that recombination, which prevents runaway increases in mutation rate, may be essential to the persistence of other viral infections that are not cleared.

Fig. 5.
Error catastrophe driven by immune response. This is a plot of the numerical solution of Model 2 in Biomedical Implications. Parameter values were taken from previous work on a similar model (47): a = 1.1, d = 0.02, β = 5, and we further assumed ...

Evolutionary Implications.

Our results suggest the possibility of a novel complement to existing explanations for why truly asexual populations are evolutionarily short-lived: Either an asexual population does not adapt and goes extinct as a result of the slow accumulation of deleterious mutations, as suggested by existing theory (18, 19, 36), or else it adapts and goes extinct as a result of the mutation-rate catastrophe. Slow accumulation of deleterious mutations is frequently cited as a likely cause for the comparatively rapid extinction of asexual organisms; in support of this idea, Muller's ratchet (19) and the related process of “mutational meltdown” (18) have been documented directly in experimental microbial populations constrained to evolve with minimal effective population sizes and high genomic mutation rates (3739). Muller's ratchet operates very slowly in large populations with low mutation rates (19, 36), and both Muller's ratchet and the mutational meltdown process can be strongly countered by adaptive substitutions (40). On the other hand, the mutation-rate catastrophe operates effectively over a large range of population sizes and is driven by adaptation. To the extent that conditions for the mutation rate catastrophe are met in real populations, then, its occurrence would greatly broaden the circumstances under which asexual populations will ultimately be less successful than sexual populations.

It is tempting to speculate that the mutation rate catastrophe phenomenon that we have observed here played a role in the early establishment of recombination in the most primitive life forms (41). It seems probable that adaptation was continual in primordial populations and that only rudimentary mechanisms of genomic proofreading and repair had evolved, such that mutation rates were closer to intolerable values than they are in most present forms. Under these circumstances, the mutation-rate catastrophe could have posed an imminent threat to any purely asexual population.

Materials and Methods

Mechanics of the Individual-Based Simulations.

At each time step, every individual produces a number of offspring, X, drawn at random from a Poisson distribution whose mean is wi/w, where wi is the fitness of the individual in the population and w = 1/NΣj=1N wj is the mean fitness of the population, i.e., X is a Poisson random variable with mean wi/w. Each offspring acquired: (i) a number XD of new deleterious mutations that were either lethal or decreased fitness by a factor (1 − MD), (ii) a number XB of new beneficial mutations that each increased fitness by a factor (1 + MB), (iii) a number XM of new mutator mutations that each increased log mutation rate by an amount MM, (iv) a number XA of new antimutator mutations that each decreased log mutation rate by an amount MA. XD, XB, XM, and XA are all Poisson random variables with means UμifD, UμifB, UμifM, and UμifA, respectively, where U is the base genomic mutation rate, μi is the relative mutation rate of individual i (this is the rate that mutates and evolves), and the f values are fractions of deleterious, beneficial, mutator, and antimutator mutations, parameters set by the user. Of the deleterious mutations produced, a fraction fL were lethal. (Values of fL as high as 0.5 did not change the extinction outcome qualitatively.) To simulate mutational effects MD ≥ 0, MB ≥ 0, MM ≥ 0, and MA ≥ 0 are continuous random variables with means mD, mB, mM, and mA, respectively. A variety of different governing distributions for these mutational effects was implemented and ranged from distributions having exponential tail probabilities to those having power-law, or very “heavy,” tail probabilities. The mutation rate ratchet operated and extinction occurred under all distributions tested.


For our simulations, the most important quantitative parameters are the fractions of deleterious and beneficial mutations relative to the genomic mutation rate, because the mutation rate catastrophe is driven by the hitchhiking of mutator alleles with beneficial mutations. Data from experimental work with E. coli provide the best available estimates of these quantities. DNA-based microbes such as E. coli appear to have a wild-type genomic mutation rate of ≈0.003 mutations per replication (42). Kibota and Lynch (43) used a mutation-accumulation experiment to estimate the rate of deleterious mutation in E. coli as 0.0002, and this suggests that the fraction of deleterious mutations in E. coli is 0.07; this value, as well as higher and lower values, were included in our simulations. There are multiple estimates of the beneficial mutation rate available in E. coli. For example, Gerrish and Lenski (44) calculated a genomic beneficial mutation rate of 2 × 10−9 per replication from stepwise fitness increases observed in experimental populations. A similar estimate (4 × 10−9) was obtained by Imhof and Schlötterer based on the dynamics of microsatellite turnover in experimental E. coli populations (45). Both of these estimates assume that the dynamics observed are a consequence of isolated beneficial mutations, but evidence is accumulating that calls this assumption into question. The dynamics of mutator hitchhiking observed in ref. 8, for example, suggest an abundance of beneficial mutations that would preclude the notion that beneficial mutations are fixed as isolated events. Indeed, a recent report by Hegreness et al. (46) suggests a much higher rate of beneficial mutation in evolving E. coli populations: ≈10−7 per replication. Overall, then, the available data from E. coli suggest that the fraction of mutations that are beneficial in adapting populations is 10−4 to 10−6. In our simulations, we varied the fraction of beneficial mutations from 10−8 to 10−3, a range that includes values wholly consistent with the available empirical data. Discussion of the remaining parameter values employed in the simulations is provided in SI Text.

Supplementary Material

Supporting Information:


We thank R. Ribeiro, H. Dahari, L. Rong, H. Y. Lee, A. M. Ponder Sutton, V. Souza, L. Eguiarte, C. Cordero, J. B. Gerrish, J. E. Fornoni, and C. P. Ferreira, R. de Boer, and two anonymous reviewers for helpful discussions and comments; and P. A. de Castro, A. C. F. Seridonio, E. R. Cazaroto, G. M. Favaro, and D. Ferreira for computer time and assistance. Portions of this work were done under the auspices of the U.S. Department of Energy under contract DE-AC52-06NA25396 and supported by National Institutes of Health Grants AI28433, RR06555, and P20-RR18754 (to A.S.P.), the National Science Foundation (P.S.), and Fundação de Amparo à Pesquisa do Estado de São Paulo (A.C.).


The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0607280104/DC1.


1. Friedberg EC, Walker GC, Siede W. DNA Repair and Mutagenesis. Washington DC: ASM Press; 1995.
2. Schaaper RM. Genetics. 1998;148:1579–1585. [PMC free article] [PubMed]
3. Sniegowski PD, Gerrish PJ, Johnson T, Shaver A. BioEssays. 2000;22:1057–1066. [PubMed]
4. Fisher RA. The Genetical Theory of Natural Selection. Oxford: Oxford Univ Press; 1930.
5. André JB, Godelle B. Genetics. 2006;172:611–626. [PMC free article] [PubMed]
6. Haldane JBS. Proc Cambridge Philos Soc. 1927;23:838–844.
7. de Visser JAGM. Microbiology. 2002;148:1247–1252. [PubMed]
8. Shaver AC, Dombrowski PG, Sweeney JY, Treis T, Zappala RM, Sniegowski PD. Genetics. 2002;162:557–566. [PMC free article] [PubMed]
9. Johnson T. Proc R Soc London Ser B. 1999;266:2389–2397. [PMC free article] [PubMed]
10. Orr HA. Genetics. 2000;155:961–968. [PMC free article] [PubMed]
11. Bull JJ, Meyers LA, Lachmann M. PLoS Comput Biol. 2005;1:450–460.
12. Wilke CO. BMC Evol Biol. 2005;5:44–52. [PMC free article] [PubMed]
13. Fijalkowska IJ, Schaaper RM. Proc Natl Acad Sci USA. 1996;93:2856–2861. [PMC free article] [PubMed]
14. Rouzine IM, Wakeley J, Coffin JM. Proc Natl Acad Sci USA. 2003;100:587–592. [PMC free article] [PubMed]
15. Tsimring LS, Levine H, Kessler DA. Phys Rev Lett. 1996;76:4440–4443. [PubMed]
16. Leigh EG. Am Nat. 1970;104:301–305.
17. Sasaki A, Nowak M. J Theor Biol. 2003;224:241–247. [PubMed]
18. Lynch M, Gabriel W. Evolution (Lawrence, Kans) 1990;44:1725–1737.
19. Haigh J. Theor Popul Biol. 1978;14:251–267. [PubMed]
20. Feldman MW, Liberman U. Proc Natl Acad Sci USA. 1986;83:4824–4827. [PMC free article] [PubMed]
21. Crotty S, Cameron CE, Andino R. Proc Natl Acad Sci USA. 2001;98:6895–6900. [PMC free article] [PubMed]
22. Eigen M. Proc Natl Acad Sci USA. 2002;99:13374–13376. [PMC free article] [PubMed]
23. Price GR. Nature. 1970;227:520–521. [PubMed]
24. Nowak M, Schuster P. J Theor Biol. 1989;137:375–395. [PubMed]
25. Takeuchi N, Hogeweg P. BMC Evol Biol. 2007;7:15. [PMC free article] [PubMed]
26. Lynch M, Blanchard J, Houle D, Kibota T, Schultz S, Vassilieva L, Willis J. Evolution (Lawrence, Kans) 1999;53:645–663.
27. Jonczyk P, Nowicka A, Iwona JF, Schaaper RM, Zygmunt C. J Bacteriol. 1998;180:1563–1566. [PMC free article] [PubMed]
28. Castro C, Arnold JJ, Cameron CE. Virus Res. 2005;107:141–149. [PubMed]
29. Bloom JD, Silberg JJ, Wilke CO, Drummond DA, Adami C, Arnold FH. Proc Natl Acad Sci USA. 2005;102:606–611. [PMC free article] [PubMed]
30. Sniegowski PD, Gerrish PJ, Lenski RE. Nature. 1997;387:703–705. [PubMed]
31. LeClerc JE, Li B, Payne WL, Cebula T. Science. 1996;274:1208–1211. [PubMed]
32. Loeb LA, Essigmann JM, Kazazi F, Zhang J, Rose KD, Mullins JI. Proc Natl Acad Sci USA. 1999;96:1492–1497. [PMC free article] [PubMed]
33. Berek C, Milstein C. Immunol Rev. 1987;96:23–41. [PubMed]
34. Berek C, Milstein C. Immunol Rev. 1998;105:5–26. [PubMed]
35. Van Valen L. Evol Theory. 1973;1:1–30.
36. Colato A, Fontanari JF. Phys Rev Lett. 2001;87:238102. [PubMed]
37. Andersson DI, Hughes D. Proc Natl Acad Sci USA. 1996;93:906–907. [PMC free article] [PubMed]
38. Zeyl C, Mizesko M, de Visser JAGM. Evolution (Lawrence, Kans) 2001;55:909–917. [PubMed]
39. Funchain P, Yeung A, Stewart JL, Lin R, Slupska MM, Miller JH. Genetics. 2000;154:959–970. [PMC free article] [PubMed]
40. Bachtrog D, Gordo I. Evolution (Lawrence, Kans) 2004;58:1403–1413. [PubMed]
41. Lehman N. J Mol Evol. 2003;56:770–777. [PubMed]
42. Drake JW, Charlesworth B, Charlesworth D, Crow JF. Genetics. 1998;148:1667–1686. [PMC free article] [PubMed]
43. Kibota TT, Lynch M. Nature. 1996;381:696–696. [PubMed]
44. Gerrish PJ, Lenski RE. Genetica. 1998;102/103:127–144. [PubMed]
45. Imhof M, Schlötterer C. Proc Natl Acad Sci USA. 2001;98:1113–1117. [PMC free article] [PubMed]
46. Hegreness M, Shoresh N, Hartl D, Kishony R. Science. 2006;311:1615–1617. [PubMed]
47. Nowak MA, May RM, Sigmund K. J Theor Biol. 1995;175:325–353. [PubMed]

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