Interfering waves of adaptation promote spatial mixing.

A fundamental problem of asexual adaptation is that beneficial substitutions are not efficiently accumulated in large populations: Beneficial mutations often go extinct because they compete with one another in going to fixation. It has been argued that such clonal interference may have led to the evolution of sex and recombination in well-mixed populations. Here, we study clonal interference, and mechanisms of its mitigation, in an evolutionary model of spatially structured populations with uniform selection pressure. Clonal interference is much more prevalent with spatial structure than without, due to the slow wave-like spread of beneficial mutations through space. We find that the adaptation speed of asexuals saturates when the linear habitat size exceeds a characteristic interference length, which becomes shorter with smaller migration and larger mutation rate. The limiting speed is proportional to μ(1/2) and μ(1/3) in linear and planar habitats, respectively, where the mutational supply μ is the product of mutation rate and local population density. This scaling and the existence of a speed limit should be amenable to experimental tests as they fall far below predicted adaptation speeds for well-mixed populations (that scale as the logarithm of population size). Finally, we show that not only recombination, but also long-range migration is a highly efficient mechanism of relaxing clonal competition in structured populations. Our conservative estimates of the interference length predict prevalent clonal interference in microbial colonies and biofilms, so clonal competition should be a strong driver of both genetic and spatial mixing in those contexts.

ONE of the most basic questions of evolutionary biology that can be studied in controlled evolution experiments is: How fast do microbial populations adapt to new environments by the accumulation of beneficial mutations? Traditionally, it was thought that the accumulation process is limited by the supply of beneficial mutations (Novick and Szilard 1950; Atwood ). If a rare beneficial mutation arises and becomes sufficiently frequent, it will expand rapidly until it is present in all individuals of the population. After the completion of such a selective sweep, the population is stationary again until the next beneficial mutation arises. The accumulation rate of beneficial mutation should thus be controlled by the appearance of new beneficial mutations, which is proportional to the population size and the mutation rate (Sniegowski and Gerrish 2010). This scenario of “periodic selection” crucially rests on the assumption that mutation rates are so small that beneficial mutations occur strictly sequentially. Evolution experiments of the last decade have shown, however, that beneficial mutation rates in microbes can be as large as 5 × 10−5 per genome per generation in the bacterium Escherichia coli with typical fitness effects in the range of a few percent (Desai ; Perfeito ; Sniegowski and Gerrish 2010). These high mutation rates have severely restricted the parameter range of periodic selection. Multiple selective sweeps, simultaneously in progress, are characteristic of a well-mixed microbial population containing >104 cells (see, e.g., estimates in ). In the past, most evolution experiments had effective population sizes exceeding this threshold, including the Lenski experiment, in which the bacterium E. coli was evolved for >50,000 generations to adapt to minimal medium (De Visser ; Miralles ; Barrick ). Accordingly, these experiments did not reveal a linear relation between adaptation speed and population size, but instead a much weaker dependence indicative of a mechanism of “diminishing returns” (De Visser ; Shaver ). The reason for this behavior is that multiple beneficial mutations arise on different genetic backgrounds simultaneously and compete with one another for sweeping through the population. As a consequence, only a small number of arising beneficial mutations can go to fixation, and most of them are lost in the competition between clones. It was first noted by Fisher (1930) and Muller (1932) that sex relaxes clonal competition and speeds up the process of adaptation as it allows beneficial mutations to be combined in a single genome even if they first appeared in different lineages. Today, this Fisher–Muller advantage of sex is one of the most important explanation for why most organisms engage in some form of genetic exchange.

In recent years, extensive research efforts have been invested in modeling clonal interference (Rouzine ; Desai and Fisher 2007; Park and Krug 2007; Hallatschek 2011) with the goal to quantitatively explain adaptation speeds and genetic diversities measured for well-mixed microbial populations (Gerrish and Lenski 1998; Desai ). Although theoretical models depend on largely unknown effect distributions of beneficial mutations, they all predict that key quantities of interest, including the speed of adaptation and the genetic diversity, depend only weakly (logarithmically) on population size and mutation rates. While these predictions shape our current view of adaptation in large asexual populations, they primarily apply to well-mixed test tube populations of microbes. Many naturally occurring microbial populations attach to surfaces in the form of biofilms exhibiting a pronounced spatial structure (Tolker-Nielsen and Molin 2000; Watnick and Kolter 2000). Even in laboratories microbes are routinely grown on agar plates by which they acquire spatial structure. More complicated spatial structure is relevant to the evolution of viral populations such as influenza and severe acute respiratory syndrome (SARS), which spread in space via human transportation networks (Hufnagel ; Eggo ). Moreover, there is increasing evidence that spatial structure is also important to the evolutionary processes involved in cancer progression (Greaves ; Merlo ; Salk ). One might thus wonder whether clonal interference is relevant to spatially extended populations and, if so, whether its effect on adaptation differs from that in well-mixed populations.

Although some simulation studies have considered the case of structured populations with one individual per deme (Gordo and Campos 2006; Gonçalves ), a crucial fact about spatially continuous populations has been ignored so far: Beneficial mutations spread in the form of waves, first described by Fisher (1937) and Kolmogorov . Since these adaptation waves spread at a constant speed, mutant clones grow linearly with time and hence much slower than they would if they grew in well-mixed populations. The slowness of selective sweeps has the consequence that growing clones are much more likely to interfere than in well-mixed populations. The link between slow adaptive waves and the potential importance of clonal interference was in fact anticipated by Fisher (1937) in his seminal article on “The wave of advance of advantageous genes.” Therein, he considers the concrete example of a mutation with selective advantage s = 1% spreading along a continuously occupied shoreline. He estimates that spreading over 100 miles might take 10,000–100,000 generations and concludes that “. . . at any one time, the number of such waves of selective advance, simultaneously in progress, must be large” (Fisher 1937, p. 367). This suggests to consider a scenario of clonal interference characterized by many interfering adaptation waves, as illustrated in Figure 1B.

Figure 1 

Simulations of asexual adaptation in a spatially extended population with absorbing boundary conditions. Stars denote the spatial location (horizontal axis) and the instant of time (vertical axis) at which new beneficial mutations become established. These events give rise to new mutant subpopulations (“clones”), which expand by waves traveling at a constant mean speed. Two drastically different adaptation scenarios are shown: (A) Periodic selection. When beneficial mutations arise sufficiently rarely, mutations sweep strictly sequentially because the waiting time for a new beneficial mutation exceeds the “fixation” time needed for a mutation to spread through the entire population. (B) Clonal interference. When the waiting time for beneficial mutations is much smaller than the fixation time, different clones compete for fixation. As a consequence, beneficial mutations are wasted (gray) unless they happen to arrive on the winning clone. Simulations were carried out with parameters L = 100, s0 = 0.25, and μ = 10−5 in A and μ = 10−4 in B; with exponentially distributed selective coefficients; and with deterministic adaptation waves.

In the following, we will use simulations and analytical arguments to answer the following key questions: What is the effect of interfering Fisher waves on the speed of adaptation and the genetic diversity in an asexual population? Are there simple mechanisms of mitigating clonal interference and thus accelerating adaptation? Are these effects relevant to microbial colonies and biofilms, and perhaps measurable in evolution experiments?

Model of Interfering Adaptation Waves

To investigate clonal interference in spatially extended populations, our model is set up to allow for an efficient simulation of many adaptation waves simultaneously in progress. The habitat is modeled by a lattice with periodic boundary conditions and is always fully populated. Each lattice site records the genetic identity of the locally dominating clone in a subpopulation. This approximate description of the gene pool of the whole population, which accelerates our simulations, is appropriate if subpopulations are strongly dominated by single clones; see and Discussion for an explicit range of validity.

We first consider a linear habitat, where the genetic state of the population is represented by a lattice of length L. Natural selection is implemented so that adaptation waves run across sites and shift the spatial extent of different clones: In each generation, the clone at site i is replaced by the neighboring clone at site j with a probability proportional to 1 + sgn(ΔW)c(|ΔW|), where c is the speed of a Fisher wave driven by a fitness difference ΔW = W − W, where the sign of ΔW defines the direction of the wave.

In effect, the replacement rule generates adaptation waves traveling at an average velocity given by c = c(ΔW). The function c(ΔW) is chosen to represent the classical Fisher wave speed (with migration rate m = ), valid for large populations with negligible genetic drift (Fisher 1937; Kolmogorov ). Alternatively, we use the wave speed given by c ∼ 2ρmΔW for strong noise (Doering ; Hallatschek and Korolev 2009) with ρm = 1, where ρ represents the population density (see for further details).

To simulate planar habitats, we represent the population on a hexagonal lattice of size L × L. Natural selection is implemented such that the clone at site i is replaced by one of the neighboring clones j with a probability proportional to its fitness, W (normalized by the fitness summed over all neighbors). This replacement rule generates adaptation waves traveling at an average velocity with a migration rate of m = , consistent with the classical Fisher–Kolmogorov wave theory (see also ).

New beneficial mutations appear at a rate μ per site (μ thus represents the product of population density ρ and beneficial mutation rate per genome) and survive genetic drift with probability 2s (Maruyama 1974), where s is the selective fitness advantage. When a mutation becomes established at position i, the fitness W of the i-th clone is updated according to W(t) = W(t − 1) ⋅ (1 + s). Effects of epistasis are absent, and the habitat is homogeneous such that selective pressures are the same throughout the habitat. The selective fitness advantage s of a mutation is chosen to be either constant s0 or drawn from a distribution with mean value s0. We choose either an exponential distribution or a hump-shaped gamma distribution with shape parameter k = 2, which are both frequently used in theoretical evolution models (Eyre-Walker and Keightley 2007). See inset in Figure 3 for the shape of these distributions.

Figure 3 

Scaling plots summarizing simulation data. The rescaled adaptation speed V/V0 is shown as a function of the rescaled mutation rate μ/μ0 (see text for the characteristic speed V0 and mutation rate μ0). For the periodic selection regime (small μ), we observe a linear relation between the adaptation and mutation rate; for clonal interference (large μ), we find the power laws V ∼ μ1/2 in linear habitats (A) and V ∼ μ1/3 in planar habitats (B). Simulations were carried out for mean selection coefficients s0 = 0.025, 0.05, and 0.1 and mutation supply rates μ = 5 × 10−6, 1.25 × 10−6, and 3.125 × 10−7. We also varied the type of adaptation wave (deterministic/noisy) and the distribution of fitness effects (black, constant selection coefficients; red, exponential distribution; blue, gamma distribution with shape parameter k = 2). Note that data for each type of fitness distributions fall on a single master curve, consistent with our scaling argument.

In our simulations of interfering Fisher waves, we vary the following parameters: the beneficial mutation rate μ per site, the habitat size L, the relation between speed c of traveling waves and fitness differences (linear habitat only), and the mean selective fitness advantage s0.

Results

Speed limit for asexuals

Starting from a population devoid of genetic variation, our model generates a population of genotypes that differ in the amount and type of beneficial mutations they carry. After a transient period, in which genetic diversity builds up, the mean fitness in the population increases at a steady pace by the fixation of beneficial mutations. We quantify the speed V of adaptation by the mean fitness increase per generation.

Figure 2 shows the adaptation speed as a function of the habitat size L for various mutation rates and selection coefficients. For small habitat sizes, the adaptation speed is linear in the habitat size, which indicates that adaptation is limited by the occurrence of mutations: Doubling the habitat size, and thus the influx of beneficial mutations, doubles the adaptation speed. We thus recover the classical regime of periodic selection, in which beneficial mutations sweep strictly sequentially, as illustrated in Figure 1A. In this regime, the dynamics of individual sweeps are irrelevant, and the mean fitness increase per generation is given by , just as in the corresponding well-mixed case. However, as the habitat size exceeds a characteristic length scale Lc, we observe that the adaptation speed begins to saturate due to clonal interference. For very large systems, the adaptation speed approaches a limiting value Vmax.

Figure 2 

The speed of adaptation in a linearly extended population. For small habitat sizes, the adaptation speed V (mean fitness increase per generation) grows linearly with the habitat size L, as expected for the periodic selection regime. As the habitat exceeds a characteristic size Lc, the adaptation speed saturates at a limiting value Vmax. Both the crossover scale and the adaptation speed limit depend on the selection advantage s0 of beneficial mutations and the mutation supply rate μ. The simulations were run assuming a constant selection coefficient and deterministic adaptation waves. The same saturation behavior is observed when selection coefficients are distributed (exponential distribution and gamma distribution). Data represent the average over ∼2000 fixation events.

These results call for an explanation of the “interference” scale Lc at which clonal interference sets in and of how the magnitude of the “speed limit” Vmax depends on the parameters, in particular mutation rates and Fisher wave speed. It turns out that—for constant fitness effects—relatively simple estimations can be given for both quantities.

The condition for clonal interference, and hence the characteristic scale Lc, can be determined by comparing two important timescales. The first timescale estimates how long it takes for a single adaptation wave to run across the length L of the habitat. This fixation time is given by tfix ≈ L/c0, where the traveling speed c0 depends on the selection coefficient s0. The second timescale is the waiting time tmut for a new beneficial mutation to become established, which is inversely proportional to both the establishment probability 2s and the rate μbL at which beneficial mutations appear. The latter depends on the dimension d of the habitat (d = 1 and d = 2 for a linear and a planar habitat, respectively). We thus obtain the mutation waiting time tmut ≈ (2s0Lμ)−1. Now, the mode of adaptation depends on the relative size of tfix and tmut: If the fixation time is smaller than the mutation waiting time (tmut < tfix), we expect periodic selection. In the opposite case, we expect clonal competition. The crossover from periodic selection to clonal interference occurs just when tfix ∼ tmut, implying a characteristic ”interference scale” of

Next, we use this estimate to find an approximate expression for the adaptation speed limit Vmax observed in Figure 2. To this end, it is convenient to express the adaptation speed in the following general form:The factor on the right-hand side is the rate at which beneficial mutations accumulate in the absence of clonal interference. The function F represents the probability that a mutation reaches fixation once established. When mutations arise sequentially, each established mutation also reaches fixation; hence F ∼ 1 for small habitats, L ≪ Lc. On the other hand, for large habitats (L ≫ Lc) clones interfere and F becomes very small. How small? Given that the adaptation speed saturates at large system sizes, as inferred from our simulation results in Figure 2, we must require that the habitat length L drops out of Equation 2 for large habitats. This can occur only if F(L/Lc) ∼ (Lc/L) for large L. Hence, we estimate the speed limit for large systems byIt can be seen from Equation 2 that the scaled adaptation speed V/V0 with V0 = s0c0/L should be a unique function of the scaled mutation rate μ/μ0 with μ0 = c0/(2s0L+1). Therefore, if we plot our data sets for different parameters in one figure using axes V/V0 and μ/μ0, they should all collapse on a single master curve. The resulting scaling plots for linear and planar habitats are shown in Figure 3. Data corresponding to constant selection coefficients (Figure 3, black line) do indeed collapse on a single master curve, even though we varied μ, s0, and the type of adaptation wave (weak/strong genetic drift). Note also the transition from a linear regime to the sublinear regimes with power law exponents and in one and two dimensions, respectively. These exponents are consistent with our prediction in Equation 3.

Moreover, Figure 3 displays results for simulations in which the fitness effects of new mutations were drawn from an exponential and a gamma distribution (with shape parameter k = 2), respectively. Note that although data for the different distributions follow our scaling predictions individually, they are slightly shifted with respect to each other. This indicates that the prefactors of the adaptation speed depend on the tails of the distribution. Broader distributions tend to yield larger adaptation speeds, apparently because they will more frequently give rise to the sampling of unusually large-s clones that outcompete average-s clones. As a consequence, fixations are more frequent and the adaptation speed is higher (see also ). The effect is stronger in the clonal interference than in the periodic selection regime.

Both the existence of a speed limit Vmax for large habitats and its dependence on mutation rates contrast with the well-mixed case, where the adaptation speed depends logarithmically on both population size and mutation rates (see Discussion).

To further characterize the clonal interference regime, we have measured fixation times and fitness correlations in linear habitats as a function of habitat size; see and . We find that tfix ∼ L3/2 for clonal interference and that the fitness variance scales linearly with the size of the habitat. Both observations suggest that our model shares some universal features with certain types of crystal growth models that have been extensively studied in physics (Kardar ).

Genetic and spatial mixing

We investigate two strategies of mitigating clonal interference and increasing adaptation speeds. The classical solution to clonal interference is recombination, which allows us to combine beneficial mutations arising on different genetic backgrounds. For well-mixed populations, it has been predicted that the adaptation speed increases with the recombination rate as V ∝ r2 for large populations where clones interfere (Neher ). This result demonstrates that increasing recombination rates can strongly speed up evolution. The effectiveness of this strategy can be appreciated from Figure 4A, which displays the adaptation speed as a function of habitat size for various recombination rates. To generate these data, we have incorporated recombination in our simulations as follows. Recombinants are produced at a rate r per site from two (haploid) individuals randomly selected from the nearest neighborhood in the parent generation (selfing is allowed). Recombinant genotypes are formed from the parental genotypes by one-point recombination: The parent chromosomes are paired and a point is randomly chosen by which both chromosomes are split. The mutations to both sides of the recombination point are then exchanged between the two chromosomes, forming the recombinant genotype (see for further details).

Figure 4 

Genetic and spatial mixing speed up adaptation. (A) Effect of recombination on the adaptation speed in a linear habitat. Simulation runs are shown with various recombination rates (see inset). Note that the adaptation speed is positively correlated with recombination rate r in the regime of clonal interference, consistent with the Fisher–Muller hypothesis. Simulations were carried out with s = const., s0 = 0.25, μ = 3.125 × 10−5, and deterministic adaptation waves. (B) Adaptation speeds vs. habitat size for varying rates ml of long-range jumps. The adaptation speed in the periodic selection regime is unchanged, but even a small rate 0.001 of long-range jumps led to fivefold increased adaptation speeds in the clonal interference regime. Simulations were carried out for s = const., s0 = 0.1, μ = 5 × 10−6, and deterministic adaptation waves.

Next, we study spatial mixing as an alternative way of mitigating clonal interference. Spatial mixing is implemented so that any given clone competes at a small rate m with a randomly chosen clone rather than with its immediate neighbor (see for details of the implementation). The effect of these long-range jumps on the adaptation speed is shown in Figure 4B. Long-range migration has no effect on the periodic selection regime, because it does not change the limiting supply of beneficial mutations. In the clonal interference regime, however, adaptation speeds are continually increasing as a function of jump rate. The effect is large: Allowing for long-range jumps only every 1000 generations approximately yields a fivefold increased adaptation speed in the clonal interference regime.

Long-range migration seems to be an efficient way to mitigate clonal interference, which has significant consequences for evolution (see Discussion). The basic mechanism can be understood as follows: Long-range migration allows clones to replicate themselves at different locations, which enables them to effectively grow at a faster rate. In other words, a clone that has led to a second “seed” by means of long-range migration will grow twice as fast as a clone with only one seed. A third seed triples the growth rate, etc. In effect, when clonal interference is absent, one obtains exponential growth with the ratewhich depends on the selection coefficient through the wave velocity relation c0 = c(s0). A derivation for this rate is given in . Note that the resulting exponential growth of mutants mirrors selective sweeps in well-mixed populations, however, with a quite different effective selection coefficient se (see also Discussion). Importantly, the resulting fast Malthusian growth accelerates the fixation process of beneficial mutations. As a consequence, mutations that would interfere without long-range migration are now more likely to arise sequentially. Consistent with this view, Figure 4 shows that the linear regime of periodic selection is extended to a larger parameter range as the long-range migration rate is increased. Of course, for sufficiently large habitat sizes, clonal interference also occurs in the presence of long-range migration. However, the number of simultaneously competing clones is reduced due to the faster sweeps. This leads to generally larger adaptation rates than for clonal interference without long-range migration, as seen in Figure 4.

Discussion

A crucial feature of adaptation in spatially extended populations is that advantageous mutations spread through the population by means of traveling waves (Fisher 1937; Kolmogorov ). This wave-like motion implies that clones grow at a constant speed, in strong contrast to the accelerating logistic growth characteristic of well-mixed populations. We have established a model of adaptation by waves to investigate the rate at which beneficial mutations are accumulated in spatially extended populations. Our model represents the gene pool of local subpopulations by the locally dominating genotype. This representation is appropriate when clonal interference is absent within these subpopulations. This sets an upper bound to the beneficial mutation rate (see ). When the beneficial mutation rate is still larger, one obtains a hybrid scenario of clonal interference on a local well-mixed scale and clonal interference by adaptation waves (E. A. Martens and O. Hallatschek, unpublished results). With the assumptions of a homogeneous habitat (constant selection pressure), short-range migration, and negligible epistasis, our model is a first step toward understanding clonal interference in spatially structured populations, such as microbial colonies or biofilms. We expect that similar phenomena to those reported here may be observed in other rapidly evolving populations. Many natural populations are, however, characterized by more complex migration patterns than the stepping-stone model used in our work, for instance, influenza or SARS (Hufnagel ; Kaluza ; Eggo ). For concrete predictions, the existing model needs to be extended to include these migration networks. There is also increasing evidence that spatial structure is relevant in evolutionary processes related to cancer progression (Salk ), suggesting that similar modeling approaches may apply (Martens ). In the following, we estimate under which conditions clonal interference might be observable in microbial colonies and discuss our key predictions and how they may be verified in microbial evolution experiments.

Clonal interference generally occurs when fixation times are larger than the waiting time for new beneficial mutations to arise in the population. Since adaptation waves travel at a constant speed and thus advance much more slowly than Malthusian sweeps, fixation times are much larger in spatially extended than in well-mixed populations. As a consequence, the regime of clonal interference is inflated in structured populations compared to well-mixed ones. The condition for clonal interference is made explicit by comparing the typical fixation time of a single beneficial mutation with the waiting time for it to arise. We found that clonal interference occurs when the habitat size is larger than a characteristic size Lc. The interference scale Lc not only defines a crossover, but also measures the typical distance an adaptation wave travels freely before it collides with another wave that arose independently of the first one. The tug-of-war between adaptation waves and the role of the interference scale Lc can be discerned from the simulation results in Figure 5.

Figure 5 

How can clonal interference and its deleterious effects on adaptation be overcome in spatially extended populations? (A) Standard clonal interference. The scale Lc is the typical distance at which adaptation waves collide in the clonal interference regime. (B) Recombination speeds up adaptation by combining beneficial variants (Fisher–Muller hypothesis). Two clones (red and blue) share interfaces (highlighted white) until recombination occurs (⊗), and a more fit recombinant arises (violet); another recombination occurs later (yellow and violet result in the green mutant). (C) An alternative mechanism of mitigating clonal interference is provided by spatial mixing, e.g., through long-range migration. Long-range jumps (arrows) accelerate selective sweeps and therefore increase both the interference scale and the fixation probability of beneficial mutations. If clonal interference is strong, spatial mixing is thus predicted to be under strong selection. Stars denote only mutations that eventually reach fixation.

Our simple quantitative expression in Equation 1 for the interference scale was found to increase with the speed of adaptation waves and decrease with the rate of beneficial mutations per site. A similar characteristic length appears in the context of soft sweeps, where μ has to be interpreted as the establishment rate for a particular beneficial mutation (Ralph and Coop 2010). For L > Lc, multiple clones carrying the same mutation collide and subdivide the habitat into patches of size Lc (see Figure 5A). This situation is similar to our case of s = const. when μ is interpreted as the occurrence rate of a particular adaptation; in our model, however, waves belonging to different adaptations keep interfering with each other until one clone has reached fixation in the habitat.

To see whether clonal interference is relevant to microbial colonies, we estimate the order of magnitude of Lc (for details see ). We assume fitness effects on the order of 1% and mutation rates in the range of 10−6–10−4 per genome and generation. Given the large variance of the thickness of biofilms and microbial colonies, we consider a range of cell densities between 1 and 1000 per square cell diameter. This yields a mutation supply rate μ that lies within the range of 10−6 and 10−1 per square cell diameter. For s = 1%, we estimate a deterministic Fisher wave speed c0 in the range of 0.1–10 cell diameters per generation by assuming that microbes in dense colonies disperse between 1 and 100 cell diameters per generation. Thus, we estimate Lc = (c0/(2s0μ))1/3 to lie within 4 and 800 cell diameters. The characteristic interference length scale Lc = (c0/(2s0μ))1/2 for a linear habitat could be relevant for expanding microcolonies. For instance, it has been shown that the expansion of a colony is driven by a thin layer of pioneer cells (Hallatschek and Nelson 2010; Nadell ). This pioneer population at the expanding front evolves in an effectively linear habitat. Applying similar estimates to the above to the context of an expanding microbial colony, we estimate an interference scale between 50 and 500 cell diameters. Importantly, the length scales both for planar and for linear habitats are much smaller than typical sizes of microbial colonies or biofilms. We thus expect clonal interference to be a widespread phenomenon—rather than the exception—in growing dense cellular clusters that conserve their spatial structure over time.

Our most important predictions for the regime of clonal interference may be summarized as follows: When the habitat size exceeds a characteristic length Lc, multiple adaptation waves are simultaneously in progress at any one time because beneficial mutations frequently arise on different genetic backgrounds. Which of the competing clones prevails and reaches fixation depends on a tug-of-war between interfering Fisher waves. Remarkably, we find that in this “frustrated” situation, increasing the habitat size does not only lead to an increased loss of beneficial mutation: The adaptation speed actually even saturates to a limiting value Vmax, which is seen for both constant and distributed selective coefficients (see ). This speed limit of adaptation and its dependence on parameters were not revealed in a previous study (Gordo and Campos 2006), seemingly because too small habitats were simulated. A simple approximate expression for the speed limit is provided by Equation 3. In particular, we found that Vmax is proportional to μ1/2 and μ1/3 in linear and planar habitats, respectively. This scaling was shown to hold for various distributions of fitness effects, including constant and exponentially distributed fitness effects, for very noisy and for deterministic Fisher waves. We may thus conclude that the predicted power laws are very robust toward details of the underlying model.

Both the speed limit and the robust power law scaling with the mutation rate are in stark contrast to well-mixed standard models, which predict that adaptation speeds never saturate and depend logarithmically on both population size and mutation rates. An experimental comparison of adaptation in well-mixed and spatially structured habitats should therefore yield qualitatively and quantitatively different results. Experiments have already found qualitative differences in the adaptation dynamics of well-mixed and structured E. coli populations (Habets , 2007; Perfeito ). Our model suggests that the speed limit of adaptation, Vmax, may be detected and thus quantified by varying the size of the microbial colony. The power law dependence of the limiting adaptation speed on both mutation rate and population density could be tested by comparing wild type with a mutator strain (Shaver ) and by manipulating the thickness of the colony, respectively.

The prevalence of clonal interference and the associated speed limit raises the question of how clonal interference may be overcome in spatially extended populations. Certainly, genetic exchange [in bacteria through lateral gene transfer (Cooper 2007)] provides a mechanism that lessens clonal interference as it allows one to bring together beneficial mutations that arose on different genetic backgrounds; see Figure 5B. This hypothesis, originally formulated by Fisher (1930) and Muller (1932), is corroborated by our simulation results in Figure 4A. However, our simulations also show that spatial mixing is a very efficient alternative mechanism of mitigating clonal interference in structured populations. Spatial mixing increases the growth rate of fitter clones and thus lowers fixation times. Lowering fixation times increases the likelihood for beneficial mutations to occur sequentially on the same background, and therefore clonal interference is attenuated. In our simulations, spatial mixing was introduced by allowing for a small rate of long-range jumps. These jumps enable clones to replicate themselves at distant locations in the habitat, which was shown to lead to accelerated clonal growth, as illustrated in Figure 5C. When clones do not interfere, the growth is exponential. The rate of this exponential growth, or effective selection coefficient se, was found to depend on both the actual selection coefficient and the rate of long-range jumps (cf. Equation 4). Importantly, we found that, due to these fast Malthusian sweeps, even very small rates of long-range jumps strongly increased adaptation speeds in the clonal interference regime (cf. Figure 4B). Thus, long-range migration seems highly beneficial in spatially extended populations with homogeneous selection pressures. Nevertheless, it remains to be seen in future work whether alleles conferring the ability of long-range migration are actually selected, even if large jumps impose a danger (for instance, a long-range jump could be fatal to a cell if the cell is swept away to a harmful environment). This evolutionary mechanism could be relevant for biofilm-forming bacteria and select for occasional switching from the biofilm state to the more efficiently dispersing planctonic phenotype.

In summary, we have analyzed a model of interfering waves. Our model suggests that clonal interference is widespread in biofilms and leads to markedly different adaptation dynamics than in well-mixed populations. In particular, the speed limit Vmax and its power law dependence on mutation rates should be quantifiable in evolving microbial colonies. Finally, we found spatial mixing to be highly beneficial in structured populations as it relaxes clonal interference and speeds up adaptation.

Table D1

Estimates of beneficial mutation rates and their selective coefficients, obtained from microbial experiments in recent literature

	U (per genome per generation)	s0	Type
Gerrish and Lenski (1998), Rozen et al. (2002), and Imhof and Schlötterer (2001)	10−9 … 10−8	—	E. coli
Elena et al. (1996)	2 × 10−9	0.03	E. coli
Desai et al. (2007)	2.4 × 10−5	0.01	S. cerevisiae
Perfeito et al. (2007)	2 × 10−5	0.02	E. coli
Sniegowski and Gerrish (2010) (data from Elena et al. 1996)	5.7 × 10−5	0.003	E. coli