- Research article
- Open Access
Long-term experimental evolution in Escherichia coli. XI. Rejection of non-transitive interactions as cause of declining rate of adaptation
© de Visser and Lenski; licensee BioMed Central Ltd. 2002
- Received: 6 September 2002
- Accepted: 30 October 2002
- Published: 30 October 2002
Experimental populations of Escherichia coli have evolved for 20,000 generations in a uniform environment. Their rate of improvement, as measured in competitions with the ancestor in that environment, has declined substantially over this period. This deceleration has been interpreted as the bacteria approaching a peak or plateau in a fitness landscape. Alternatively, this deceleration might be caused by non-transitive competitive interactions, in particular such that the measured advantage of later genotypes relative to earlier ones would be greater if they competed directly.
To distinguish these two hypotheses, we performed a large set of competitions using one of the evolved lines. Twenty-one samples obtained at 1,000-generation intervals each competed against five genetically marked clones isolated at 5,000-generation intervals, with three-fold replication. The pattern of relative fitness values for these 315 pairwise competitions was compared with expectations under transitive and non-transitive models, the latter structured to produce the observed deceleration in fitness relative to the ancestor. In general, the relative fitness of later and earlier generations measured by direct competition agrees well with the fitness inferred from separately competing each against the ancestor. These data thus support the transitive model.
Non-transitive competitive interactions were not a major feature of evolution in this population. Instead, the pronounced deceleration in its rate of fitness improvement indicates that the population early on incorporated most of those mutations that provided the greatest gains, and subsequently relied on beneficial mutations that were fewer in number, smaller in effect, or both.
- Relative Fitness
- Beneficial Mutation
- Fitness Improvement
- Transitive Model
- Fitness Gain
In this paper, we seek to test two hypotheses that could, in principle, account for this deceleration. (H1) The observed deceleration indicates that the rate of evolutionary adaptation is truly slowing down. This occurs as the evolving population approaches a fitness peak or plateau because the remaining number of beneficial mutations, their marginal effect, or both become progressively smaller. This hypothesis assumes that fitness values of the chronologically ordered population samples are both qualitatively and quantitatively transitive; i.e., they follow a strict competitive hierarchy, such that the cumulative fitness improvement relative to the ancestor could be predicted from the incremental gains over the constituent time intervals. (H2) The rate of adaptation continues at the initial rapid pace, and the apparent deceleration is an artifact of using the ancestor as the common yardstick to measure adaptation. Under this hypothesis, fitness values are non-transitive. Consequently, the total fitness improvement relative to the ancestor cannot be predicted from the incremental gains over smaller intervals.
Several studies have documented non-transitive interactions. Paquin and Adams  observed unexpected declines in fitness in populations of Saccharomyces cerevisiae evolving in glucose-limited chemostats, when fitness was based on competition with the ancestral strain. However, when a series of chronologically ordered isolates was each competed against its immediate predecessor, the later isolate was invariably more fit. The precise mechanism responsible for this non-transitivity was not identified, but presumably involved the accumulation of metabolic by-products in the culture media, which affected competition. Rainey and Travisano  observed non-transitive fitness interactions between three different variants of Pseudomonas fluorescens that evolved in cultures without physical shaking, which allowed gradients or other heterogeneities to develop. The non-transitive effects were based on competitive trade-offs between niche specialists, which provided an advantage to each type when it was rare relative to one or both other types in the overall heterogeneous environment. Non-transitive interactions sometimes also result from cyclical competitive hierarchies that resemble the "rock-paper-scissors" game, in which rock crushes scissors, scissors cut paper, and paper covers rock . Biological examples of such cyclical hierarchies include: (i) competition among toxin producers, resistant types, and sensitive non-producers in species that use allelopathy to compete, such as coral-reef dwelling invertebrates  and various microbes [8, 9]; (ii) competition between three morphologically different types with distinct reproductive strategies in side-blotched lizards ; and (iii) other situations with more than two limiting resources that can lead to complex dynamics in plankton communities .
In this study, we seek to test whether non-transitive interactions could account for the observed deceleration in fitness gains, measured relative to the ancestral strain, in the long-term E. coli evolution experiment. To that end, we first develop transitive and non-transitive models that can explain – both equally well – the measured trajectory of fitness relative to the ancestor, but which make quantitatively different predictions about the fitness values that would result from competing evolved samples taken at different time points. We then perform experiments to test which model is better.
Transitive and non-transitive models
According to the transitive model, the fitness of two samples relative to one another can be calculated from the fitness of each sample relative to their common ancestor: for example, W(C:B) = W(C:A) / W(B:A). If the same samples were all to compete against a more fit clone isolated from later in the experiment (instead of against the ancestor), then their relative fitness values should shift down by a constant proportion, as shown in Figure 2A. When these same values are ln-transformed, the trajectories against a set of progressively more and more fit competitors form a series of parallel lines (Figure 2B). Also, the fitness trajectories against a set of competitors isolated at equal intervals from the evolution experiment should be more closely packed against the later competitors, which are progressively more similar in their fitness relative to one another.
In general, non-transitive competition models can take many different forms. All such models will have the property that ln-transformed fitness trajectories measured against a series of competitors isolated from different times deviate from parallelism. More specifically, we are interested in the possibility that non-transitive interactions might produce the observed deceleration in fitness relative to the ancestor, yet be consistent with a constant rate of on-going adaptive change. Figure 2C shows the non-transitive model that uniquely fulfils both these conditions. A ln-transformation of this model does not produce parallel fitness trajectories, nor do these trajectories become more closely packed when competitions are performed against later competitors (Figure 2D). The transformation does show, however, that all of the trajectories have equal and maximum slopes, which correspond to samples that are temporally adjacent.
Fitness trajectories measured against five different competitors
Several important features of these data are immediately apparent. First, the fitness trajectories against all five competitors decelerate over time, confirming the findings reported previously based on competitions against the common ancestor only [1–3]. Second, the five ln-transformed fitness trajectories are nearly parallel to one another. Third, the trajectories are much more closely packed against the clonal competitors from later generations. All these features are consistent with the transitive model, but not with the non-transitive model. To formalize this conclusion, we perform two rigorous statistical tests in the next section.
Statistical tests support transitive model, reject non-transitive model
ANOVA testing for non-transitive competitive interactions. Sample refers to the 21 population samples taken from generations 0, 1,000, 2,000, ..., and 20,000. Clone refers to the 5 neutrally marked clones from generations 0, 5,000, 10,000, 15,000, and 20,000. Each sample competed against each clone with 3-fold replication. The sample-by-clone interaction is used to test for any deviations from the transitive model.
We tested two hypotheses for the observed deceleration of fitness improvement in an experimental population of evolving E. coli. One hypothesis assumes competitive interactions are transitive, and that the deceleration indicates an actual decline in the rate of adaptation – owing to fewer beneficial mutations, mutations having smaller benefits, or both – which is expected when a population approaches a fitness peak or plateau. The other hypothesis assumes that the instantaneous rate of adaptation is as fast after 20,000 generations as was the initial rate, and that the observed deceleration is an artifact resulting from non-transitive interactions. By measuring the fitness of samples obtained from the population over the course of 20,000 generations relative to five clones from different time points, we found absolutely no evidence for non-transitive interactions using two statistical tests. First, an analysis of variance showed that the fitness trajectories relative to clones from different time points were almost perfectly parallel, as predicted by the transitive model, whereas the non-transitive model predicts strongly non-parallel trajectories. The interaction term in this analysis tested for any possible deviation from transitivity, and not only the particular form that could have accounted for the deceleration in fitness gains measured relative to the ancestor. Second, the cumulative fitness increase after 20,000 generations, relative to the ancestor, was consistent with the incremental gains measured across a series of temporal intervals, as predicted by the transitive model but significantly contrary to the non-transitive model.
Our purpose in presenting these findings is not to argue that non-transitivity is either uninteresting or unimportant in some other systems. As reviewed in the Background, non-transitive interactions have been well documented in several systems. Also, the features of the environment used in the E. coli long-term experiment (including fairly low glucose concentration and the absence of physical structure) were chosen, in part, to minimize this potential complication, in order to focus on the most fundamental dynamics of evolution. We do not even claim that all forms of frequency-dependent interaction (of which non-transitivity is only one type) are completely absent from the long-term evolution experiment with E. coli. In fact, it has been shown elsewhere that another population than the one we have studied here has a balanced polymorphism that is maintained by a strong frequency-dependent interaction , and other populations contain clones with much weaker advantages when rare .
What we do assert, based on our new data and analyses, is that non-transitive effects are either absent or much too weak to account for the pronounced overall deceleration in the rate of fitness gains measured relative to the ancestor. Consequently, this deceleration indicates that the pace of genetic adaptation has indeed slowed, although it has not completely stopped , in the long-term E. coli evolution experiment. Therefore, this and the other evolving populations must have fairly quickly incorporated mutations that provided the greatest gains, and their slower subsequent adaptation reflects beneficial mutations that are fewer in number, smaller in effect, or both [1, 14].
Bacteria and culture conditions
The bacteria used in this study were all derived from a single population, designated Ara-1, that is part of the long-term evolution experiment [1–3]. Details concerning the founding strain of E. coli B, culture conditions, and so on are given elsewhere . In brief, this population was propagated serially by 1:100 daily transfers for 20,000 generations (= 3,000 days) in Davis minimal (DM) salts medium supplemented with glucose at 25 μg/mL and incubated at 37°C. In this study, we use population samples from 1,000-generation intervals, which were stored at -80°C with glycerol as a cryo-protectant. Each sample contains millions of cells and includes essentially all of the genetic diversity present in the population at the time of storage.
The Ara-1 population and its ancestor are unable to grow on the sugar L-arabinose. However, an Ara+ variant of the ancestor has been generated, and it has previously served as the common competitor for samples from the evolving Ara-1 population. Numerous experiments have shown that this Ara+ marker is selectively neutral under the culture conditions used in the long-term evolution experiment. However, this marker allows evolved and ancestral competitors to be distinguished and enumerated by plating on tetrazolium-arabinose (TA) indicator agar .
For this study, we needed Ara+ clones derived from the Ara-1 population at generations 5,000, 10,000, 15,000 and 20,000 in addition to the ancestral Ara+ clone. We used an Ara+ mutant clone that was isolated previously from a clone sampled at generation 10,000 from Ara-1 . For generations 5,000, 15,000 and 20,000 we randomly chose single clones from each of the corresponding samples taken from Ara-1. Each clone was grown to high density, concentrated by centrifugation, and plated on minimal-arabinose agar medium  to find spontaneous Ara+ mutants. Three Ara+ mutants were isolated from each clone, and each mutant was assayed in preliminary competitions against the population sample from which it came. We retained for this study the single Ara+ mutant whose estimated fitness relative to its source sample was closest to unity (and, in all cases, not significantly different from 1). To summarize, this study uses 21 population samples of Ara- cells taken at 0, 1,000, 2,000, ..., 20,000 generations from the evolving Ara-1 population, and 5 neutrally marked Ara+ mutant clones derived from generations 0, 5,000, 10,000, 15,000 and 20,000 of the same population.
Measuring relative fitness
We performed a total of 315 competitions to measure the fitness of each population sample relative to each marked clone, with three fold-replication. The competitions followed the same protocol described in detail elsewhere . To summarize briefly, the competitions were performed under the exact same culture conditions as in the long-term evolution experiment. To ensure that any two competitors were comparably acclimated to the competition environment, they were simultaneously removed from the freezer, grown separately in a nutrient-rich broth for one day, and then acclimated for another day to the competition environment. The competitors were then mixed at a 1:1 volumetric ratio, diluted 1:100 into fresh DM and incubated at 37°C for one day. Appropriate dilutions were plated on TA agar at the start of the competition and again after 24 h, to estimate the initial and final numbers of each competitor. In a few cases, plate counts were less than 50 total or less than 15 for one competitor. In these cases, we repeated the competition experiments to obtain more accurate estimates of relative fitness based on higher counts. The relative fitness of two competitors was calculated simply as the ratio of their realized population growth rates during competition . Let N(0) and N(1) denote initial and final population densities, respectively, and let subscripts i and j indicate the two competitors. Then relative fitness is calculated as:
W(j:i) = ln [Nj(1) / Nj(0)] / ln [Ni(1) / Ni(0)].
Relative fitness is dimensionless because the same time units are used in calculating both competitors' realized population growth rates.
Standard statistical analyses were used throughout, with one exception. The jackknife method [, pp 795–799] was used to calculate the confidence interval around the mean value in Figure 5B. In particular, the mean cumulative gain observed over four 5,000-generation intervals was calculated as the product of four successive means, each based on six measured values (three from a later sample competed against an earlier clone, and three from a later clone competed against an earlier sample). Owing to the non-linear dependence of a multiplicative product of means on the measured values, one cannot calculate a confidence interval using standard methods. The jackknife method yields a confidence interval that accurately reflects variation in the 24 underlying measurements. The jackknife method was not needed to compute the confidence intervals around the means in Figure 5A and 5C, because both of these means are simple arithmetic averages and hence do not involve any non-linearities.
The authors thank T. Czárán and R. Hoekstra for helpful discussions and two anonymous reviewers for suggestions that improved the manuscript. JAGMdV's research is supported by the Netherlands Organization for Scientific Research. REL's long-term evolution experiment is funded by the United States National Science Foundation (DEB-9981397).
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