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The impact of population size on the evolution of asexual microbes on smooth versus rugged fitness landscapes
© Handel and Rozen; licensee BioMed Central Ltd. 2009
- Received: 1 October 2008
- Accepted: 18 September 2009
- Published: 18 September 2009
It is commonly thought that large asexual populations evolve more rapidly than smaller ones, due to their increased rate of beneficial mutations. Less clear is how population size influences the level of fitness an asexual population can attain. Here, we simulate the evolution of bacteria in repeated serial passage experiments to explore how features such as fitness landscape ruggedness, the size of the mutational target under selection, and the mutation supply rate, interact to affect the evolution of microbial populations of different sizes.
We find that if the fitness landscape has many local peaks, there can be a trade-off between the rate of adaptation and the potential to reach high fitness peaks. This result derives from the fact that whereas large populations evolve mostly deterministically and often become trapped on local fitness peaks, smaller populations can follow more stochastic evolutionary paths and thus locate higher fitness peaks. We also find that the target size of adaptation and the mutation rate interact with population size to influence the trade-off between rate of adaptation and final fitness.
Our study suggests that the optimal population size for adaptation depends on the details of the environment and on the importance of either the ability to evolve rapidly or to reach high fitness levels.
- Fitness Landscape
- Fitness Peak
- Beneficial Mutation
- Asexual Population
- Final Fitness
Understanding the factors that influence the evolution of microbial populations not only provides fundamental insights into evolutionary processes [1–4], but is also of considerable applied importance, owing to the fact that many microbes are pathogenic. Development of a predictive framework of microbial evolutionary dynamics is central to understanding processes such as the evolution of drug resistance [5–7] and the emergence of novel infectious diseases [8, 9]. Numerous interacting factors determine evolutionary patterns of microbes, but all are likely influenced by the size of the microbial population. In this work we focus our attention on the consequences of population size in asexual microbes, and study how changes in this parameter interact with other factors to modify its role and importance in adaptive dynamics.
On the other hand, a small population will generate only a subset of all possible 1-step beneficial mutations, with few mutations that confer large fitness effects [13, 15, 16]. Both the reduced supply rate of new mutations and their smaller fitness benefits contribute to the expected slower rate of adaptation of small versus large populations. However, at the same time the small populations will follow more stochastic adaptive trajectories , and this increases their ability to explore the more distant fitness landscape. With this broader exploration comes an increased likelihood of reaching more distant and higher fitness peaks (Figure 1 bottom). Thus while both large and small populations can become trapped upon local optima, small populations may be more able to avoid this trap and consequently reach higher fitness peaks. In the present work, we use computer simulations to explore this phenomenon in more detail, focusing on factors that might modify the role of population size during adaptive evolution. In particular, we focus on the interaction of population size with factors that are likely to influence the adaptive trajectories of microbes; namely ruggedness of the fitness landscape, the target size of adaptation and mutation rates.
We simulate the evolution of bacteria as they undergo repeated cycles of growth and serial dilution [1, 14]. At the start of each simulation, the population consists of N0 identical clones. The bacteria go through D rounds of division, and each bacterium produces offspring depending on fitness, f, as 2 f . After D divisions, serial transfer, modeled as multinomial sampling, reduces the population size back to N0 which initiates another round of exponential growth. This procedure is iterated until the desired number of generations is reached. Because bacterial death is ignored, the only way a given clone can be eliminated is if it is not sampled during serial transfer [17, 18].
By adjusting the rules for how the L-mutant neighborhood is chosen, we can tune the fitness landscape from one that is completely smooth to one that is completely rugged. For the smooth landscape, each newly created mutant is assigned a mutant neighborhood that is identical to that of the ancestral strain. In other words, the mutation does not alter the fitness effects of any subsequent mutations that might be obtained. Under these conditions there is a single fitness peak. At the other extreme, for a completely rugged landscape, every new mutant is assigned an entirely new L-mutant neighborhood with values for the fitness effect of each new mutation re-sampled from p(s). This means that a new mutation changes the fitness effects of all other possible mutations. In this scenario there is no correlation between the fitness effects of the L-mutants from a parent clone and those available to its mutant offspring. By changing the fraction, r, of the L sites that are replaced, we can tune the amount of ruggedness of the landscape from smooth (r = 0) to completely rugged (r = 1). By considering a broad range of values for r and L, we can explore a range of scenarios in order to identify conditions where changes in parameters lead to qualitative changes in adaptation. Figure 2 schematically shows an example for r = 0.6.
initial size of population
102, 104, 106
size of mutant neighborhood (number of accessible 1-step mutants)
5, 50, 500
fraction of mutant neighborhood that is changed (ruggedness of fitness landscape)
0.1, 0.5, 1
beneficial mutation rate per replication
number of divisions per growth cycle
distribution of fitness effects
Adaptation on a rugged landscape
On a smooth fitness landscape, all populations will eventually reach the sole fitness peak, with the larger populations doing so more rapidly. However, this can change during adaptation on a rugged landscape, as explained above. Here, large populations are expected to evolve almost deterministically. This allows them to quickly reach the highest local fitness peak, where, if asexual, they can become trapped. In contrast, a smaller population size allows for more stochastic trajectories on the fitness landscape, and this can occasionally lead to higher fitness peaks. The transition from more stochastic to more deterministic trajectories occurs as the mutation supply rate, S, becomes so large that a population is able to completely sample all possible 1-step beneficial mutations, i.e. if S ≈ L . The mutation supply rate is the product of mutation rate and effective population size, S = N e μ. For the three initial population sizes we consider here, N0 = 102, 104 and 106, an effective population size given by N e ≈ DN0 , and mutation rate μ = 10-6, the mutation supply rates are S s = 0.001, S m = 0.1 and S l = 10 for the small, medium and large populations respectively. We initially choose the size of the 1-step neighborhood to be L = 50, which means S l ≈ L, S m <L and S s « L. Thus we expect the large population to evolve mostly deterministically, while the medium population is expected to evolve somewhat slower, but with the potential of reaching higher fitness peaks. Because the small populations have S s « 1, they are expected to operate in the strong selection weak mutation limit, where evolution will be slow because it is limited by the infrequent creation of beneficial mutations [35, 36].
The impact of landscape ruggedness
Rank/CV for different population sizes.
L = 50, r = 1
L = 50, r = 0.5
L = 50, r = 0.1
Changing mutant neighborhood
L = 5, r = 1
L = 50, r = 1
L = 500, r = 1
Changing mutation rate
μ = 0.5/N e
The impact of the size of the mutant neighborhood
For instance for L = 5, the medium populations have a reduced amount of stochasticity and are more likely to have reached a (local) peak, compared with the L = 50 situation (see Rank and CV in Table 2). This results in a lower fraction of populations that reach fitness higher than that of the large population (compare Figure 7 top row L = 5 with L = 50). For small L, the small populations are less disadvantaged in terms of adaptive "speed" and are able to more frequently, although still quite rarely overall, reach higher fitness peaks than larger populations (compare Figure 7 bottom row L = 5 with L = 50).
For the large (L = 500) scenario, evolution for the large population becomes markedly more stochastic (see CV in Table 2), leading to a broader exploration of the fitness landscape. This results in less frequent instances where the medium populations reach higher fitness than the large populations (compare Figure 7 top row L = 50 with L = 500). This supports the intuitive understanding that if more beneficial mutations are accessible, the population size that optimizes the trade-off between the speed of adaptation and the magnitude of the adaptive response shifts towards larger populations. Indeed, in the limit of L → ∞, every clone has access to all possible other mutants, in essence reducing the system to a smooth landscape on which the large populations are always favored [30, 34].
Mutation rates versus population sizes
Above, we explained how the relation of the mutation supply rate, S, and the mutant neighborhood, L, are relevant for determining whether adaptation tends to be dominated by stochastic or deterministic change. The mutation supply rate is the product of population size and mutation rate. It is known that population size and mutation rate can have differential effects on the evolutionary dynamics [13, 40, 41]. For example, fixation times are faster in smaller populations, even though mutations arise less often.
It is generally accepted that large populations will tend to evolve more rapidly than smaller ones. This is caused by two related factors. First large populations have an increased supply of beneficial mutations each generation, which decreases the waiting time for new advantageous mutations. Second, large populations have increased access to mutations that confer large benefits. These factors imply that larger populations gain an advantage by taking larger adaptive steps during population evolution. However, as we have shown in a previous study , sometimes smaller populations can reach higher levels of fitness. Here, we have explored this phenomenon in more detail. We found that while large populations evolve faster on both smooth and rugged landscapes, on the latter there can be a trade-off between speed and the potential to reach high fitness peaks. Because large populations tend to fix the most advantageous mutations first and thereby follow a very limited set of adaptive trajectories, they have a tendency to become trapped on local fitness peaks. In contrast, smaller populations become fixed for a wider range of possible beneficial mutations which leads to increased variation in adaptive trajectories across populations and allows some populations to avoid becoming trapped on local peaks. However, the potential to reach higher fitness peaks can come at the cost of a slower speed in adaptation. The optimal population size therefore likely depends on the relative importance of speed versus final fitness.
We further showed that for a rather smooth fitness landscape, there is no advantage in following more stochastic adaptive trajectories; however, even an intermediate amount of ruggedness can be sufficient to occasionally favor more stochastically evolving populations of smaller size. Experimental studies suggest that at least some amount of ruggedness is present in natural situations [38, 44–47]. We also showed that when the size of the mutational target under selection is very large or very small, the system converges to an effectively smooth landscape where large populations are favored.
Lastly, we found some evidence that for a fixed mutation supply rate, small populations evolved more rapidly and more stochastically, which allowed them to reach higher fitness compared to larger populations in a majority of simulations. We suggest that this can be attributed to clonal interference acting in larger populations, which limits the amount of within population variation and can retard the rate of adaptation . To keep the mutation supply rate constant, it was necessary to increase the mutation rate for the small populations. That this tended to confer an advantage may imply that small populations, such as bacterial pathogens at or following the bottleneck during transmission, may benefit by adopting a transient mutator phenotype in order to successfully colonize new hosts. An important caveat to this is that if the mutation load increases with mutation rate, with an associated increase in genetic drift during bottleneck transmission, a mutator strategy would carry a profound cost, both for individual populations and descendant lineages in separate hosts .
Although medium and small populations can exceed the fitness of larger populations, we note that this outcome does not occur in all, or even most, simulations. More important, the degree to which this result is realized is highly dependent upon underlying landscape architecture. For example, as is most clearly evident in Figure 5, there is considerable variation in the fraction of cases where populations of medium size exceed the fitness of large populations, with a broad range from 0.02 to 0.54. Several features of the fitness landscape influence the potential outcome of the adaptive walks. First, if the closest fitness peak is a global peak, medium and small populations would fail to capitalize on their greater searching ability. This would also apply if the local peak is the highest peak within a certain "radius" of the starting location in the fitness landscape, since a far away peak might never be reached by any of the populations. Second, the difference between the global peak and accessible local peaks may be negligible, in which case differences in adaptive magnitude across populations of different sizes will be similarly small. Finally, the global peak may not be accessible at all, in which case the smaller populations will again fail to capitalize upon their potential search advantages.
As with any model, we have made several simplifications. For instance we excluded death of bacteria and only allowed the loss of novel mutants to occur through stochastic loss during sampling via serial dilution. The inclusion of stochastic drift  would likely not change the bulk of our results, but it might impact some of the details, especially for our small population size with N0 = 100.
A second simplification is our exclusive focus on asexual populations. A number of studies have shown that the incorporation of recombination can help to overcome clonal interference or can help populations to more easily escape from local fitness peaks [50–53], though recombination might not be always beneficial . Extending our model to allow for recombination is a focus of future studies and will allow us to understand how recombination may help large populations to avoid becoming trapped upon local fitness peaks.
We used our simulation to study populations that ranged in size over 4 orders of magnitude. In this range, we found that our large populations exhibited clonal interference and very rarely escaped from local fitness peaks. However, a number of recent studies suggest that if the population size is large enough, the impact of clonal interference might be reduced [43, 55–57]. Additionally, very large populations are expected to more easily escape from local fitness peaks [58–62]. For the combination of population size and severity of bottleneck we used in our simulations, we found that deleterious mutants were removed from the population most of the time before they could reach appreciable frequencies and lead to compensatory mutations. This may suggest that for evolution through growth-bottleneck cycles (which applies not only to laboratory situations, but is likely also applicable to many pathogens), the bottleneck size interacts strongly with the population size and other parameters to determine the dynamics of the evolutionary process [17, 18, 63]. Further investigation of the interactions of population size, landscape ruggedness and mutation rate with bottleneck size, and the importance of different types of mutations during growth-bottleneck cycles  deserves further study.
In summary, we have shown that for asexual populations evolving on rugged fitness landscapes, there can be a trade-off between speed of adaptation and the attainable fitness, which strongly depends on the underlying fitness landscape. This suggests that the optimal population size likely depends on both the details of the fitness landscape and the relative importance of speed versus final fitness.
We thank Matthew Bennett and Arjan de Visser for comments on an earlier version of the manuscript. Daniel Rozen was supported by funds from the University of Manchester.
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