Our multiple-mutation analysis predicts that the speed of evolution and the variation in fitness within a population both increase logarithmically with N
and with Ub
. It also shows that neglecting multiple mutations is a serious flaw of one-by-one clonal interference analyses. We set out to test these predictions by evolving asexual populations of diploid budding yeast in glucose-limited media for 500 generations at three different effective population sizes ranging from 1400 to 3.5 × 106
, each with two different mutation rates: “nonmutator” populations and msh
2Δ “mutator” populations with Ub
estimated to be 10 times higher [36
] (based on the elevation of mutation rate at two particular loci, and hence only a rough estimate; see Methods). The generation time in this media was initially about 130 minutes, compared to 90 minutes in rich media.
We periodically measured the fitness of each entire population by mixing a sample of it with a derivative of the ancestral strain that had been labelled with yellow fluorescent protein, growing the mixed population for 20 generations, and determining the ratio of the two strains at the beginning and end of the assay by using flow cytometry to distinguish labelled from unlabelled cells (see Methods). The total fitness changes over the 500 generations were used to obtain the average speed of evolution (). We also measured the distribution of fitnesses within some of the evolved populations by isolating 96 individuals from each and then measuring their individual fitnesses ().
Figure 2 (a) Average speed of adaptation (ν) of our experimental populations, in percent fitness increase per generation, versus ln[NUb] (scaled to the smallest nonmutator population). N is the effective population size (which takes on three values, N (more ...)
Figure 3 Fitness distributions of evolved populations after 500 generations, for three independently evolved lines of the: (a) largest mutator populations, (b) smallest mutator populations, (c) largest nonmutator populations, (d) smallest nonmutator populations. (more ...)
Our data are clearly inconsistent with the simple successional fixation prediction, ν linear in NUb (p < 0.001). Other simple interpretations are ruled out by the observed time dependence of the mean fitness of our populations (). The rate of fitness increase is roughly constant, in particular showing no evidence of slowing down as the experiment progresses (if anything, a slight speeding up is seen). This indicates that neither antagonistic epistasis (i.e. the combined effect of two beneficial mutations being less than the sum of their separate effects) nor a limited supply of beneficial mutations (i.e. “running out” of beneficial mutations) can be responsible for the observed weak dependence of ν on N. Note that the batch culture environment remains the same throughout our experiment, and the populations are in exponential phase throughout, so environmental changes cannot explain these results.
Our data are also inconsistent with one-by-one clonal interference analyses, because of their assumption that mutations fix singly in succession. This inconsistency is most apparent for our largest populations. Our large nonmutator populations increased in fitness by about 4 to 7 percent in 500 generations. This is not enough time for two or more mutations adding up to 4 to 7 percent to fix one by one (i.e. successionally). For example, two 3.5 percent effect mutations would take a minimum of 1000 generations to fix successionally; all other combinations adding to between 4 and 7 percent would take even longer. A similar argument applies to our large mutator populations.
Thus if single beneficial mutations fix successionally, one
large such mutation must be responsible for almost the entire observed fitness increase. However, this is also inconsistent with the data. shows that the mean fitness of our populations increases smoothly, and the individual profiles are similar to each other. Both features imply that the evolution is not dominated by single large mutations. If it were, the mean fitness would remain constant for a time and then rapidly increase by the amount of the large mutation. For a 7 percent mutation, for example, most of the increase in fitness would occur in just 30 generations (fixation times are much longer because mutants spend a long time while rare). The fitnesses of different populations would also show a wide range of kinetics depending on whether their large effect mutations occurred early or late (). Yet this is not at all what we see. Instead, the gradual increase in fitness and similar kinetics between lines strongly suggest that many smaller mutations are steadily accumulating (). This cannot happen unless multiple mutations sweep together: successional sweeps of small effect mutations would take far too long (). A more detailed discussion, including other inconsistencies with one-by-one clonal interference and special circumstances in which one-by-one clonal interference could produce the observed results, is presented in the supplemental material
Figure 4 Computer simulations showing the kinetics of the increase in mean fitness in 10 simulated populations of size N = 3.5 × 106, the same as our large experimental populations. (a) Assuming a single 7% effect mutation is responsible for the evolution (more ...)
The above arguments suggest that the multiple-mutation analysis is the correct explanation for our results. A key qualitative prediction of this analysis is that the width of the fitness distributions in large populations should be greater than in small populations. In contrast, one-by-one clonal interference predicts that fitness distributions will show pronounced fluctuations over time for any population size: narrow and dominated by a single clone most of the time or, if measured during a selective sweep, clearly bimodal. Similar behavior would arise from a simple successional fixation (one-locus) analysis, except that the rate of sweeps would increase dramatically in large populations. In actuality, for both mutators and nonmutators, we find that the fitness distributions of large populations are broader than of small ones ().
These predictions can be made quantitative: the expected widths of the fitness distributions and the speeds of evolution are given by the formulas in Box 1
. These predictions depend on just two unknown parameters: the typical size of the beneficial mutations responsible for the fitness increase,
, and the rate at which these beneficial mutations occur, Ub
. One might worry that for any experimental data, there would be a combination of Ub
that would produce a good fit. This is not so. In the smallest populations, NUb
is so small that they can only be in the successional fixation regime. These populations tightly constrain Ub
in a way that is independent of the multiple-mutation theory, ruling out arbitrary Ub
which might have yielded good fits to the other data. Within these constraints, we fit Ub
from the data, yielding values Ub
= 2.4 × 10−4
for mutator populations (hence Ub
= 2.4 × 10−5
for nonmutators), and
= 2 percent. Details of the theory-independent constraints and the fit to data are described in the Methods. The resulting comparison between theory and experiment is summarized in Figs. and . The predicted increases in mean fitness (which give the speeds of evolution shown in ) and widths of the fitness distributions are each within a single fitness increment
of the experiments — as accurately as theory could possibly predict. There are, however, small systematic discrepancies: the theory overestimates the mean fitness increases for mutator populations and underestimates their width, while making the opposite errors in nonmutators. This is likely due to deleterious mutations, which we now consider.
Deleterious mutations complicate the shapes of the fitness distributions. However, their effects are most pronounced on the less-fit side of the distributions: on the more-fit side, all the clones are depleted similarly by deleterious mutations and the modifications of the shape of the distribution are small. Thus in the analysis described above we only use the more-fit side, above the median. But deleterious mutations may indirectly a ect the more-fit side of the fitness distributions, for example by decreasing the median fitness. They will also cause a reduction in the mean fitness of the population and hence could reduce the observed speed of evolution.
Unfortunately, the effects of deleterious mutations depend on the unknown distribution of their fitness decrements, so precise predictions are impossible. We can, however, estimate their maximum impact by looking at the small population fitness distributions. The small nonmutator fitness distributions are no wider to the right than expected from the measured experimental errors (), which implies that deleterious mutations do not significantly reduce the mean fitness in nonmutators, nor do they a ect the above-median fitness distribution width. In other words, the contribution of deleterious mutations in nonmutators is minor and limited to a slight increase in the width of the less-fit tail. In mutators, on the other hand, the width to the right of the median in the small populations could be entirely due to reduction of the median by deleterious mutations, entirely due to beneficial mutations, or due to some combination of the two. This means that in all the mutator populations, deleterious mutations may decrease the mean fitness by at most two percent and broaden their fitness distributions by convolving them (defined in [37
]) with a distribution of standard deviation 1.4 percent. These shifts lead to slight changes in the best-fit Ub
which lead to the opposite shifts in the predicted results for nonmutators. These corrections roughly account for the systematic discrepancies between experiments and the multiple-mutation theory.
Although the multiple-mutation picture better explains our experiments, clonal interference must nevertheless also occur. Mutations of very small effect are certainly being regularly wasted, and this process partially determines the typical size,
, of the mutations that dominate the evolution. However, our data indicates that the effect omitted in one-by-one clonal interference analyses, the accumulation of multiple mutations, is crucial. After fitting
from data to implicitly account for clonal interference effects, the simple multiple-mutation theory is consistent with our experiments, especially once we consider the additional effects of deleterious mutations.
Several other recent experimental studies have also found that, as in our experiments, the speed of adaptation increases less than linearly with population size and mutation rate [4
]. This has been taken as support for one-by-one clonal interference. But our multiple-mutations analysis also predicts a specific form of this less than linear dependence on N
, albeit for different reasons. This earlier work is not suffciently detailed to distinguish between one-by-one clonal interference and our multiple mutations model.
If the beneficial mutations of size
≈ 2% are point mutations, combining the estimate of Ub
with the per base pair mutation rate of order 10−9
per generation [38
] suggests that the target size for beneficial mutations in our experiments is a few thousand base pairs. This is substantially higher than the beneficial mutation rates in several earlier studies done in different environments [39
], but closer to recent estimates by Joseph and Hall [41
]. It is possible that there are several targets of roughly a hundred base pairs, such as genes where inactivating one of the two copies in a diploid conveys an advantage, or a number of much smaller mutational hot spots (as found by [42
]), whose mutation rate is much higher than the average per base pair mutation rate — perhaps having evolved to allow rapid mutational switches between different metabolic environments encountered in the natural history of budding yeast.
Finally, we note that the logarithmic increase in the speed of evolution with N and Ub in the large-NUb multiple-mutations regime is dramatically slower than the linear successional mutations regime. The difference has many implications. For example, the potential advantage of sex in combining mutations from different lineages becomes more pronounced in large populations, while mutator phenotypes become less useful as population sizes increase.