Abstract
Periodic bottlenecks in population sizes are common in natural (e.g., host-to-host transfer of pathogens) and laboratory populations of asexual microbes (e.g., experimental evolution) and play a major role in shaping the adaptive dynamics in such systems. Existing theory predicts that for any given bottleneck size (N0) and number of generations between bottlenecks (g), populations with similar harmonic mean size (HM=N0g) will have similar extent of adaptation (EoA). We test this widely cited claim using long-term evolution in Escherichia coli populations and computer simulations. We show that, contrary to the predictions of the extant theory, HM fails to predict and explain EoA. Although larger values of g allow populations to arrive at superior benefits by entailing increased number of individuals, they also lead to lower EoA. We also show analytically how the extant theory overestimates the effective population size relevant for adaptation. Altering the current theory using these insights, we propose and demonstrate that N0/g (and not N0g) successfully predicts EoA. Our results call for a re-evaluation of the role of population size in two decades of microbial population genetics and experimental evolution studies. These results are also helpful in predicting microbial adaptation, which has important evolutionary, epidemiological and economic implications.
Introduction
Population size is a key demographic parameter that affects several ecological and evolutionary processes including the rate of adaptation (Gerrish and Lenski, 1998; Lanfear et al., 2014; Wilke, 2004; Desai et al., 2007; Samani and Bell, 2010), efficiency of selection (Petit and Barbadilla, 2009), organismal complexity (LaBar and Adami, 2016), fitness decline (Katju et al., 2015), repeatability of evolution (Lachapelle et al., 2015; Szendro et al., 2013), etc. Interestingly though, what constitutes a proper measure of population size often depends on the ecological/evolutionary question being addressed (Charlesworth, 2009). For example, for an ecologist studying population-dynamics, the total number of individuals is often the appropriate metric (Dey and Joshi, 2006). On the other hand, for a conservation biologist studying the loss of heterozygosity (Ellstrand and Elam, 1993), or an evolutionary biologist who wishes to predict how much a population would adapt over a given time relative to the ancestral state (i.e., extent of adaptation or EoA) (Samani and Bell, 2010), the effective population size (Ne) might be a more suitable measure (Campos and Wahl, 2009; Charlesworth, 2009; Desai et al., 2007; Wahl and Gerrish, 2001). Consequently, it is crucial to use the relevant measure of population size while constructing or empirically validating any evolutionary theory.
Experimental evolution using asexual microbes has been one of the key tools in validating several tenets of evolutionary theory (reviewed in (Kawecki et al., 2012)). Most such studies deal with populations that face regular and periodic bottlenecks during their propagation (Kawecki et al., 2012). Since the absolute population size keeps changing regularly in such experiments, the harmonic mean population size (HM) is often estimated as the ‘effective population size’ in such studies (Lenski et al., 1991; Wahl and Gerrish, 2001; Campos and Wahl, 2010). Specifically, if a population grows from size N0 to Nf via binary fissions within a growth phase, and is diluted back periodically to N0 by random sampling at the end of the growth phase, then the effective population size is given by Ne ≈ N0*log2(Nf/N0) = N0g, where g refers to the number of generations between successive bottlenecks and N0g is the harmonic mean size (Lenski et al., 1991). Other measures of the adaptively relevant population size used in experimental evolution studies are conceptually similar, and are of the form Ne = N0gC, where C is a constant (Desai et al., 2007; Wahl and Gerrish, 2001; Campos and Wahl, 2009; Samani and Bell, 2010).
Several experimental studies, employing a variety of asexual model organisms, have used HM for quantifying the effective population size (Desai et al., 2007; Samani and Bell, 2010; Lenski et al., 1991; Raynes et al., 2012, 2014; De Visser and Rozen, 2005; Rozen et al., 2008). However, there is no direct empirical validation of the suitability of HM as a measure of population size that can explain the EoA. More critically, recent findings have questioned the validity of HM as the evolutionarily relevant measure of population size in both asexual (Raynes et al., 2014) and sexual (Jiménez-Mena et al., 2016) organisms. Here we address this issue using a combination of agent-based simulations and long-term evolutionary experiments using Escherichia coli. We first test the suitability of HM as a predictor of EoA, and show that both real bacterial as well as simulated, populations with similar values of N0g can have markedly different adaptive trajectories. Secondly, we demonstrate that although increasing the value of g promotes adaptation through an increased supply of variation, it also impedes adaptation by restricting the spread of beneficial mutations, brought about by reduced efficiency of selection. Thus, the resultant EoA is an interplay between these two opposing aspects of g and contrary to the extant theoretical expectations (Campos and Wahl, 2009; Heffernan and Wahl, 2002), EoA has a negative relationship with g. Thirdly, we show that populations with similar HM can not only have different fitness trajectories, but can also differ markedly in terms of how frequency- distribution of fitness amongst individuals changes during adaptation. We then show that, for a given mutation rate, N0/g (we call this quantity the adaptive size, AS) is a much better predictor of EoA trajectories, i.e., populations with similar AS have similar fitness trajectories and populations with higher AS adapt faster. Finally, we demonstrate that during adaptation, populations with similar AS can converge on similar trajectories of EoA using mutations with widely different fitness effects. Our findings challenge the current notion of how population size influences adaptation.
Materials and Methods
Experimental evolution
Selection regimens
We propagated three distinct regimens (LL, SL, and SS) of Escherichia coli MG 1655 populations for more than 380 generations. 8 independently evolving replicate lines each of LL (large HM and large Nf, selection in flasks; culture volume: 100ml), SL (small HM but large Nf, selection in flasks; culture volume: 100ml), and SS (small HM and small Nf, selection in 24-well plates; culture volume: 1.5 ml) were derived from a single Escherichia coli K-12 MG1655 colony and propagated in Nutrient Broth with a fixed concentration of an antibiotic cocktail containing a mixture of three antibiotics at sub-lethal concentrations (See Supplementary Methods). The three population regimens experienced different numbers of evolutionary generations (g) between periodic bottlenecks (i.e., before they were sub-cultured). SS and SL had similar HM (i.e., N0g) albeit obtained through different combinations of N0 (SS>SL) and g (SL>SS) such that the Nf of SL was approximately 73 times larger than that of SS. The Nf of SL was similar to that of LL, while the harmonic mean size of LL was > 16,500 times larger than that of SL and SS (Table S1). LL was bottlenecked 1/10 every 12 hours, SS was bottlenecked 1/104 every 24 hours, and SL was bottlenecked 1/106 every 36 hours. 1 ml cryostocks belonging to each of the twenty four independently evolving populations were stored periodically.
Fitness assays: To reconstruct the evolutionary trajectories of our experimental bacterial populations, we measured bacterial growth using an automated multi-well plate reader (Synergy HT, BIOTEK ® Winooski, VT, USA). Bacterial growth was measured in the same environment that the populations experienced during evolution using OD at 600 nm as a proxy for population density. Bacteria from the cryostocks belonging to each of the 24 populations were grown in 96 well plates. Each cryostock-derived population was assayed in three measurement-replicate wells in a 96 well plate. Each well contained 180 μl growth medium containing 1:104 diluted cryostock. The plate was incubated at 37ᴼC, and shaken continuously by the plate-reader throughout the growth assay. OD readings taken every 20 minutes during this incubation resulted in sigmoidal growth curves. Fitness measurements were done using cryostocks belonging to multiple time-points in order to reconstruct evolutionary trajectories. While making trajectories, it was made sure that every 96 well-plate contained populations belonging to similar time-points (in terms of number of generations). We used the carrying capacity (K) and maximum population-wide growth rate (R) as the measure of fitness (Novak et al., 2006). K of a population was defined as the maximum OD value attained over a period of twenty four hours (the highest value in the sigmoidal growth curve) while R was estimated as the maximum slope of the growth curve over a running window of four OD readings (each window spanning one hour) (Karve et al., 2015; Ketola et al., 2013; Vogwill et al., 2016; Lachapelle et al., 2015). Fitness measurements were done using cryostocks belonging to multiple time-points in order to reconstruct evolutionary trajectories. While making trajectories, it was made sure that every 96 well-plate contained populations belonging to similar time-points (in terms of number of generations).
Statistics: Bacterial fitness was analyzed for each of the two growth parameters (K and R) using a nested- design ANOVA with population regimen-type (SS, SL or LL) as a fixed factor and replicate-line (1-8, nested in population-type) as a random factor. We corrected for the error derived from multiple tests using Holm-Šidàk correction (Abdi, 2010). Since we observed adaptive trait trajectories with curves of diminishing returns throughout our study, we used extent of adaptation (EoA) at any given time to compare the three regimens. Throughout this study, EoA refers to the amount of fitness gained with respect to the ancestor.
Simulations of microbial evolution
We simulated fission-based asexual population growth under resource limited conditions to further investigate the issue and generalize our results. In our model, an individual bacterium was characterized by three principal parameters: efficiency, threshold, and body-mass. The simulation began with a fixed amount of resources available in the environment, utilized by the bacteria for growth. A typical individual was represented by an array (coded in the C programming language) that specified three principal parameters: (1) Bodymass, (2) Efficiency, and (3) Threshold. Bacteria consumed resources in an iterative and density-dependent manner. The parameter Bodymassi of a typical individual (say individual i) represented how big the particular individual is during a given iteration. Its efficiency (K_effi) specified how much food it assimilated per iteration. If population size/ K_effi < 1, 10*(1 - (population size/ K_effi)) units were added to Bodymassi. Otherwise, Bodymassi remained unchanged. Bodymassi increased with cumulative assimilation. The moment Bodymassi becomes greater than or equal to thresi (its threshold parameter), the individual i underwent binary fission and divided into two equally sized daughter individuals. Each fission event had a fixed probability of giving rise to mutations based on a mutation rate that remained constant for all individuals in the population. K_effi and thresi mutate independently, and were the only two parameters that could undergo mutation. The mutated value was drawn from a static normal distribution with the frequency of deleterious mutations being much higher than that of beneficial mutations, which is in line with experimental observations (Kassen and Bataillon, 2006; Eyre-Walker and Keightley, 2007). The distribution of mutational effects remained fixed throughout the simulation (Kassen and Bataillon, 2006) due to which, EoA was expected to eventually approach a plateau. When the population ran out of resources (once the amount of body-mass accumulated per unit time by the population went below a pre-decided threshold so that the sigmoidal curve reached a plateau), it was sampled according to the sampling ratio being studied. The above process was repeated for 400 generations, where each generation represented two-fold growth in population size (see Supplementary Methods for a detailed description of the model).
Results
HM failed to predict and explain the EoA trajectories of experimental populations
HM failed to explain the EoA trajectories of experimental populations. In spite of having similar values of N0g, the SL and SS regimens had markedly different adaptive (EoA) trajectories for K (Fig. 1a; See Table S3 for the p-values) as well as R (Fig. 1b; Table S4). This observation is consistent with recent empirical findings that question the validity of harmonic mean as the effective population size (Raynes et al., 2014). Surprisingly, SS had a larger overall EoA than SL despite having lower Nf. This suggests that bottleneck intensities might have a greater effect on EoA trajectories than absolute population sizes. We grew LL as a control regimen in order to address whether Nf itself could predict EoA trajectories. Despite having Nf similar to SL, LL typically had much larger EoA than SL.
Simulations also revealed that HM fails to explain and predict adaptive trajectories
To obtain greater and generalizable insights into the various determinants of EoA trajectories, we used an Individual Based Model (IBM) with different values of N0 and g, such that the product (N0g) remained similar. If N0g were a good predictor of how much a population is expected to adapt, then these three treatments were expected to show similar EoA (Campos and Wahl, 2009; Wahl and Gerrish, 2001). This was not found to be the case for both K (Fig. 2a) and R (Fig. 2b), which was consistent with our experimental observations of EoA trends in SL and SS (Fig. 1).XX’, SS’, and SL’ were also found to be remarkably different in terms of the adaptive increase in average efficiency of individuals (Fig. S4a). We also found that populations with similar harmonic mean sizes could differ remarkably in terms of the frequency distributions of the efficiency parameters amongst their constituent individuals (Fig. S5). In order to elucidate why N0g could not explain EoA trajectories, we determined how EoA varied with N0 and g, independently.
EoA varied positively with N0 but negatively with g
If N0g were a good measure of the adaptation effective population size (i.e., the measure of population size which has a positive relationship with EoA and can explain EoA trajectories), then increasing either or both of N0 and g should lead to greater EoA. We tested this intuitive prediction via simulations using several combinations of N0 and g. Although EoA was found to increase with greater N0 (Fig. S6a and S6b), the relationship between EoA and g turned out to be negative (Fig. 3; Fig. S6c and S6d). The latter result implied that large values of Nf impeded adaptation in populations even when the population size during the bottleneck (N0) was held constant. The nature (sign) of this relationship between EoA and g was found to be robust to changes in mutation rate over a 100-fold range in our simulations (Fig. S7).
A negative relationship between EoA and g is particularly surprising because, in populations with similar N0, increase in g is expected to lead to an increase in the available variation. This is because a larger value of g automatically means an increase in Nf with a concomitant increase in the number of fissions per evolutionary generation (and hence chances of mutation). All else being equal, this should have led to greater EoA. Since that was not the case, we went on to check if these slowly adapting populations (with similar N0 but higher g values) were limited by the availability of variation, both qualitatively and quantitatively.
The availability of beneficial traits could not explain why EoA varied negatively with g
Consider SM1 and SM4, treatment regimens which had similar starting population size (N0) after the first bottleneck but had g values of 3.32 and 13.28 respectively (SM refers to sampling ratio, expressed in terms of log(10) (see Fig. 4 and 5). SM1 grew to a final size of 10N0 in one growth phase (i.e., before bottleneck), while SM4 grew to 104N0. In other words, SM1 faced a periodic bottleneck of 1/10 whereas SM4 was sampled 1/104 periodically. Since SM4 experienced approximately 279 times more fission events than SM1 per evolutionary generation, the former was expected to undergo more mutations and consequently show more variation. Moreover, SM4 was also expected to arrive at very large-effect benefits that were so rare that the probability of SM1 stumbling upon them was vanishingly low due to its lower mutational supply. As expected, SM4 was not found to be limited by the supply of variation as it had a consistently higher within- population coefficient of variation in terms of efficiency values than SM1 (Fig. 4). SM4 also had a continual access to highly fit genotypes (Fig. 5a) that were inaccessible to SM1 throughout the simulations. On the basis of these observations, EoA can be expected to vary positively with g and thus SM4 was expected to be fitter than SM1 at a given point of time in general. However, counterintuitively, SM4 had a consistently lower EoA than SM1 (Fig. 4). Evidently, harsher periodic sampling impeded adaptation despite resulting in increased substrate for selection. We also found that although higher census size allowed SM4 to arrive at extremely rare mutations with very large benefits, these mutations failed to survive the harsh periodic bottlenecks by rising to large enough frequencies (Fig. S8). This explains why arriving at these rare mutations with very large benefits did not make SM4 adapt more than SM1 in a sustained manner. However, this does not explain why the EoA in SM4 was consistently lower than that of SM1.
The negative relationship between EoA and g can be explained in terms of efficiency of selection
EoA depends on an interplay between two factors: (I) generation of beneficial variation and (II) an increase in the frequency of beneficial variants as an interaction between selection and drift. The first depends upon the supply rate of beneficial mutations (Sniegowski and Gerrish, 2010), and, as shown above, the relative availability of beneficial mutations across SM1 and SM4 does not explain why SM4 was adaptively inferior to SM1. An increase in the frequency of beneficial variants is aided by the efficiency of selection (in eliminating deleterious mutations and spreading beneficial ones), which is reflected by how quickly the modal phenotype of a population approaches its best phenotype. Since all our simulations were started with a symmetric (uniform) distribution of efficiency and threshold amongst individuals, directional selection was expected to give rise to a negatively skewed distribution of efficiency. In such negatively skewed distributions, the smaller the difference between the mode and the mean, the higher would be the efficiency of selection. Selection operated more efficiently in SM1 than in SM4 as the modal phenotypic class converged with the best available phenotypic class in most SM1 populations (as reflected by the string of zeros in SM1 (Fig. 5b)), but failed to do so in all SM4 populations. Moreover, the distance between the location of the modal class and the mean class was much smaller in SM1 as compared to SM4 (Fig. 5c).
N0/g is a better predictor of EoA than N0g
Since our simulations suggested that the rate of adaptation is positively related to N0 and negatively related to g, we went on to test if N0/g is a better predictor of adaptive trajectories than N0g. N0/g indeed turned out to be a better predictor of EoA trajectories not only in our simulations (Fig. 2, 6, and S9), but also for our experiments. The N0/g values of LL, SS and SL populations were approximately 3.01*109, 1.13*104, and 5.02*103, respectively, which led to a predicted EoA trend of LL>SS>SL, which was observed for both K and R in the experiments (Fig. 1). We call the quantity N0/g the adaptive size (AS) and propose that AS should be used to make predictions about EoA in periodically bottlenecked asexual populations. We also found that populations with similar AS can have markedly different trait distributions at any given time despite having very similar trajectories of mean fitness (Fig. S10). Evidently, similar distributions of EoA-affecting traits amongst individuals imply similar mean EoA trajectories, but the converse is not true. We elaborate on this result in the discussion section.
Discussion
Periodic bottlenecks lead to increased variation but reduced adaptation
The growth of many natural asexual populations is punctuated by episodic bottlenecks caused by, for example, abrupt dissociation from hosts or spread of infections across hosts (reviewed in (Abel et al., 2015)), etc. Moreover, periodic sampling during sub-culturing is a common feature of most asexual populations propagated during experimental evolution studies (Kawecki et al., 2012; Lenski et al., 1991). Therefore it is important to appreciate the complex role played by periodic bottlenecks in such populations. Most experimental evolution studies with asexual microbes are started with either genetically uniform/clonal replicate populations or a relatively small inoculum. Thus, the generation and survival of de novo beneficial variation is the principal basis of adaptation in such populations (Kawecki et al., 2012; Barrick et al., 2009). Populations that experience more binary fissions per generation are expected to generate more de novo beneficial variation and thus, to have a higher extent of adaptation. The number of binary fissions per generation is given by N0*(2g-1)/g (see below), which varies positively with g (Fig. S11). Therefore, if EoA depends solely upon the amount of variation generated by mutations, then all else being equal, EoA is expected to vary positively with N0 and g, which is consistent with the expectation that HM (≈N0g) should be a good measure of the adaptive effective population size. However, this line of reasoning disregards the loss of variation during periodic bottlenecks, which increases in intensity with increasing g due to decrease in the fraction of the population being sampled. It has been predicted that the probability that a beneficial mutation of a given size survives a bottleneck varies negatively with the harshness of sampling (i.e., increasing g) (Wahl et al., 2002). However, since the overall rate of adaptation depends upon the product of beneficial mutational supply rate and survival probability, it has also been suggested that bottlenecked populations may adapt faster than populations of constant size (Wahl et al., 2002). Heffernan and Wahl have proposed that exponential growth is a more potent evolutionary force than abrupt periodic bottlenecks, and increasing g increases the probability of fixation of a beneficial mutation (Heffernan and Wahl, 2002). The sign of the relationship between EoA and g has not been put to empirical test yet, but, as shown above, the extant formula (HM) for the adaptive effective size implies a positive relationship (Campos and Wahl, 2009; Wahl and Gerrish, 2001). Our experiments (Fig. 1) and simulations (Fig. 2) did not support this prediction and EoA was found to have a negative relationship with g (Fig. 3).
In order to explain this discrepancy, we simulated populations with similar values of N0 (i.e., bottleneck size) but different degrees of harshness of the bottlenecks, namely SM1 (lenient bottleneck, g =3.32) and SM4 (harsh bottleneck, g = 13.28) (Fig. 4 and Fig. 5). The high fitness phenotypes had a higher probability of getting lost due to the harsh sampling in SM4 (1 in104) than in SM1 (1 in 101) as reflected in Fig. S8. Moreover, asexual reproduction prevents multiple alternative beneficial mutations from coming together in any given individual. Therefore, alternative beneficial mutations compete with each other for fixation. This competition, also known as clonal interference (CI), impedes the speed of increase in average population-wide fitness (Gerrish and Lenski, 1998; Wilke, 2004; Park and Krug, 2007; Sniegowski and Gerrish, 2010). This is because increasing the availability of beneficial mutations beyond a particular level does not result in a concomitant increase in adaptation rate, thus leading to a relationship of diminishing returns between adaptation-rate and beneficial mutational supply (Gerrish and Lenski, 1998). Since the intensity of CI varies positively with the number of such competing mutations (Gerrish and Lenski, 1998), the effects of CI would be much more pronounced in SM4 populations which have ~279-times greater number of mutations per generation compared to the SM1 populations.
N0/g determines the amount of variation that ends up surviving the bottleneck
If binary fission is the basis of exponential growth from N0 to Nf (one growth phase), the number of fissions is given by Nf - N0. The number of rounds of fissions that take place during this growth phase is log2(Nf /N0), which is equal to g. Therefore, the number of new mutations that occur during this growth phase (from N0 to Nf) is given by µ*N0*(2g – 1) where µ is the mutation rate. At the end of the growth phase, the population is bottlenecked by random sampling of N0 individuals. Ignoring the arrival times and fitness differences across mutations and plugging in Nf/N0=2g, the number of new mutations that would putatively end up surviving this sampling from Nf individuals to N0 individuals would then be given by (N0/Nf)*µ*N0*(2g – 1) = (2-g)* µ*N0*(2g – 1). If 2g >> 1 (i.e., if g is large), then 2g – 1 ≈ 2g and (N0/Nf)*µ*N0*(2g – 1) ≈ µ*N0. Populations that face different bottleneck ratios undergo different number of bottlenecks (and growth phases) in a given number of generations. For example, a population that faces a periodic bottleneck of 1/104 undergoes 30 growth phases in 400 generations whereas a population that faces a periodic bottleneck of 1/10 undergoes 120 growth phases in the same number of generations. Therefore, in order to compare different populations, at a given point of time, in terms of the amount of variation that survives sampling, we need to calibrate this quantity with g, the number of generations per growth phase. Since the growth phase from N0 to Nf spans g evolutionary generations, the number of new mutations created per generation that would end up surviving the bottleneck would be given by µ*N0/g. We acknowledge that this is a simplification and in reality, both arrival times and mutational competition are significant factors that shape evolutionary trajectories (Sniegowski and Gerrish, 2010) (see below for further discussion).
Populations with remarkably different beneficial mutations can show similar EoA
We emphasize that N0/g can be a good predictor of mean adaptive trajectories (Fig. 6) but not necessarily of the trait-distributions (Fig. S10). In other words, populations with markedly different absolute sizes but similar N0/g can use beneficial mutations of different effect sizes to arrive at similar mean EoA values in a given amount of time (Fig. S10). This explains how populations that are different in terms of absolute sizes can show similar EoA trajectories. Since fixation probabilities associated with individual mutations determine how trait distributions change over time during adaptation (Heffernan and Wahl, 2002; Patwa and Wahl, 2008), the above results also suggest that knowing the fixation probabilities may not enable one to predict EoA trajectories.
Conventional measures have overestimated the effective population size for adaptation
Our findings have major implications for comparing results across experimental evolution studies. Adaptive dynamics in asexuals are highly influenced by the beneficial mutation supply rate in the population (UbNe), where Ub is the rate of spontaneous occurrence of beneficial mutations per individual per generation and Ne is the effective population size (reviewed in (Sniegowski and Gerrish, 2010)). In the context of a given environment, it can be assumed that Ub is a constant fraction (k) of µ, such that Ub = kµ. As shown above, µN0/g is an approximate measure of the number of new variants created per generation that are expected to survive bottlenecking (if the arrival times of mutations and competition across mutations are ignored). Therefore it is expected that the quantity kµN0/g would reflect the beneficial mutational supply per generation. Therefore, by definition, kµ(N0/g) ≈ UbNe, which implies that Ne ≈ N0/g (since Ub= kµ). Unfortunately, N0/g is an overestimate of Ne because µN0/g overestimates the number of new variants created per unit time by ignoring the arrival times of mutations and mutational competition. However, since N0 g is g2 times larger than N0/g, and g typically varies between 3 and 20 in most experimental evolution studies (Kawecki et al., 2012), it is clear that the traditional formula for HM can overestimate the adaptive effective population size by 1 to 2 orders of magnitude. Moreover, since the number of competing beneficial mutations per generation (a measure of the intensity of clonal interference) varies positively with Ne (Sniegowski and Gerrish, 2010), our study highlights that the conventional formula also overestimates the extent of clonal interference in periodically bottlenecked populations which can potentially complicate the interpretation of empirical studies on this topic (Desai et al., 2007). Furthermore, our results can be used to explain some of the previously observed discrepancies in terms of adaptive effective population sizes in experimental evolution studies. For example, a recent study found that three experimental asexual populations with similar values of N0g could show significantly different evolutionary dynamics (Raynes et al., 2014). Our study suggests that the observed differences in the evolutionary outcomes might be explained by the fact that these populations differed remarkably from each other in terms of N0/g. We also propose that AS (=N0/g) should be used to compare different studies in terms of the reported average speed or extent of adaptation in meta-analyses of laboratory evolution of asexual populations. We resolve the adaptively relevant size of such populations into two components (N0 and g) based on their effects on EoA. Since our study demonstrates how the relationship of EoA with N0 is opposite to its relationship with g, these results should be useful in predicting how much adaptive change can be expected from different experimental designs. For example, decisions on culture volumes (well-plates versus flasks) and dilution ratios in laboratory evolution can be made on the basis of the above results to best suit the demands of the experiment.
Evolution of carrying capacity can feedback into adaptive trajectories
Finally, we point out that both our experiments and simulations demonstrate that carrying capacity (K) can evolve during adaptation in asexual microbes (Fig. 1a and 2a). Most models of microbial adaptation do not take into account such adaptive changes in carrying capacity (Gerrish and Lenski, 1998; Desai et al., 2007; Wahl and Gerrish, 2001; Campos and Wahl, 2010) despite there being clear empirical evidence that carrying capacity can change during adaptation (Novak et al., 2006). Moreover, if the carrying capacity itself changes during the experiment, the constancy of bottleneck ratio (unchanging value of g) ensures that N0 also changes concomitantly as the population evolves. This means that the periodicity of bottlenecks introduces a positive feedback during evolution if K increases adaptively – larger value of N0 would make a population evolve higher K, which in turn would increase the next N0, and so on. We think that this aspect of fitness should not be omitted from theoretical models of how microbes evolve, particularly under resource limited conditions, which are a common feature of experimental evolution protocols (Kawecki et al., 2012; Lenski et al., 1991).
Author contributions
Y.D.C. and S.D. conceived and designed the study. Y.D.C. conducted the experiments. S.D., S.I.A., and Y.D.C. developed the model. S.I.A. wrote the model-code. Y.D.C. ran the simulations. Y.D.C. and S.D. wrote the paper.
Supplementary Data
Statistical analysis of empirical results
Agreement between experiments and simulations
The results of our experiments and simulations agree well in terms of the range and dynamics of adaptation over identical time-scales in numerically similar populations. This applies to both measures of population-level fitness: carrying capacity (K) and maximum growth rate (R).
Qualitative differences in EoA trajectory-shapes brought about by large differences in population size
As expected (Sniegowski and Gerrish, 2010), very small populations showed staicase-like (stepwise) trajectories of fitness increase.
Changes in standard deviation with sample size
Since our simulations are agent-based (and consequently take a very long time to run), we decided to operate on a sample size of 8 replicates per population type throughout our study.
Adaptation in terms of efficiency and threshold
Multiple measures of fitness in our study revealed that harmonic mean is not a good predictor of adaptive trajectories because populations with similar harmonic mean size can have markedly different adaptive trajectories (Fig. S4 and Fig. 2 (Main-text)). Identical trends were observed when such populations (XX’, SS’, and SL’) were compared in terms of two different measures of population level fitness (Fig. 2 (Main-text)). In terms of fitness at the level of individuals, efficiency showed the same trend as R and K (Figu. S4a). However, the adaptive trajectories corresponding to XX’, SS’, and SL’ were almost identical when expressed in terms of threshold. Threshold evolved (decreased) so quickly and to such a large extent in almost all population types that we simulated in this study (regardless of their HM) that most populations had similar trajectories of threshold decrease (also see Fig. S6d). Consequently, despite threshold being an important determinant of fitness, adaptive differences amongst populations were best expressed and explained in terms of trajectories of increase in efficiency and not in terms of decrease in threshold. The trends shown by adaptive trajectories of efficiency increase were identical to those shown by adaptive trajectories of K and R. Due to the above reasons, we focussed on population-wide trait distributions only in terms of efficiency.
Adaptive changes in distributions of efficiency in populations with similar HM
Our simulations revealed that populations with similar harmonic mean size can differ appreciably from each other not only in terms of their adaptive trajectories but also in terms of how the distribution of fitness amongst their constituent individuals changes during adaptation.
Relationship of EoA with N0 and g
As predicted by the extant theory (Wahl and Gerrish, 2001; Campos and Wahl, 2009), the extent of adaptation (EoA) had a positive but saturating relationship with N0. However, we found that EoA varied negatively with g. Populations with similar N0 but different g had markedly different adaptive trajectories (Fig. S6C and Fig. 3, 4 (Main-text)).
Relationship between EoA and g at three different mutation rates
Differences in the locations of the best class before and after bottleneck
N0/g is a better predictor of EoA trajectories than N0g
Populations with similar N0/g had remarkably similar adaptive trajectories in terms of both efficiency and threshold (Fig. S9). These populations had similar adaptive trajectories despite differing in terms of the intensity of the periodic botleneck over a 1000-fold range. While N0*g failed to predict adaptive trajectories over this bottleneck range (Fig. 2 and 6 (Main text)), N0/g could act as a much better predictor of adaptive trajectories.
Population with similar mean EoA trajectories can differ remarkably in terms of distributions of the corresponding fitness-affecting trait
Poulations that have similar mean adaptive trajectories can nevertheless have remarkably different distribution of fitness amongst their constituent individuals, and can also differ in terms of how these distributions themselves change over time. LBbar and HB have markedly different distributions of fitness amongst their constituent individuals during the course of adaptation, despite having similar fitness trajectories.
Acknowledgements
We thank Milind Watve, M.S. Madhusudhan, Shraddha Karve and Sachit Daniel for their invaluable suggestions and insightful discussions. We thank Amitabh Joshi for critical comments on an earlier draft of the manuscript. Y.D.C. was supported by a Senior Research Fellowship from Indian Institute of Science Education and Research, Pune. S.I.A. thanks Department of Science and Technology, Government of India for financial support through a KVPY fellowship. This project was supported by an external grant from Department of Biotechnology, Government of India and internal funding from Indian Institute of Science Education and Research, Pune.
Footnotes
Conflict of interest: The authors do not have any conflict of interest.