Non-selective effects, like genetic drift, are an important factor in modern conceptions of evolution, and have been extensively studied for constant population sizes (Kimura, 1955; Otto and Whitlock, 1997). Here, we consider non-selective evolution in the case of growing populations that are of small size and have varying trait compositions (e.g. after a population bottleneck). We find that, in these conditions, populations never fixate to a trait, but tend to a random limit composition, and that the distribution of compositions “freezes” to a steady state. This final state is crucially influenced by the initial conditions. We obtain these findings from a combined theoretical and experimental approach, using multiple mixed subpopulations of two Pseudomonas putida strains in non-selective growth conditions (Matthijs et al, 2009) as model system. The experimental results for the population dynamics match the theoretical predictions based on the Pólya urn model (Eggenberger and Pólya, 1923) for all analyzed parameter regimes. In summary, we show that exponential growth stops genetic drift. This result contrasts with previous theoretical analyses of non-selective evolution (e.g. genetic drift), which investigated how traits spread and eventually take over populations (fixate) (Kimura, 1955; Otto and Whitlock, 1997). Moreover, our work highlights how deeply growth influences non-selective evolution, and how it plays a key role in maintaining genetic variability. Consequently, it is of particular importance in life-cycles models (Melbinger et al, 2010; Cremer et al, 2011; Cremer et al, 2012) of periodically shrinking and expanding populations.

Funding: EF was funded by the Deutsche Forschungsgemeinschaft, Priority Programme 1617 "Phenotypic heterogeneity and sociobiology of bacterial populations", grant FR 850/11-1 and the German Excellence Initiative via the programme “Nanosystems Initiative Munich” (NIM). HJ was funded by the Deutsche Forschungsgemeinschaft, Priority Programme 1617 "Phenotypic heterogeneity and sociobiology of bacterial populations", grant JU 333/5-1 ( http://www.spp1617.org/ ). KW was supported by the Elitenetzwerk Bayern. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Stochastic effects play an important role in population dynamics [ 8 – 11 ], particularly when competition between individuals is non-selective. Most previous theoretical analyses have studied how a non-selectively evolving trait can spread and eventually replace all other variants (fixate) under conditions in which the population size remains constant [ 2 , 12 , 13 ]. However, both natural and laboratory populations frequently experience exponential growth. Here we show that genetic diversity in growing populations is maintained despite demographic noise, and reaches a stationary but random limit. We used a well-controlled model system in which well-mixed co-cultures of a wild-type Pseudomonas putida strain and an isogenic mutant were grown under non-selective conditions. Multiple subpopulations were generated, each containing a random number of individuals of each strain. Depending on the average initial population size and the strain ratio, we observed distinct stationary probability distributions for their genetic composition. Moreover, we showed that the dynamics of growing populations can be mapped to Pólya urn models [ 4 ], permitting the observed maintenance of genetic diversity to be understood as the random limit property of a fair game between individual strains. Generalizing the Pólya urn model to include the effects of random initial sampling and exponential growth allowed us to predict the evolution of the composition distribution. Using numerical and analytical methods we found that the distribution broadens at first but quickly “freezes” to a stationary distribution, which agrees with the experimental findings. Our results provide new insights into the role of demographic noise in growing populations.

Results and Discussion

Evolutionary dynamics is driven by the complex interplay between selective and non-selective (or neutral) effects. The paradigm of non-selective evolution originates from the seminal work of Kimura [1], in which he solved the Wright-Fisher model, thus showing that non-selective effects—and specifically genetic drift—can have a determinant role in evolution. His results sparked an ongoing debate about the nature and potency of randomness as a fundamental evolutionary force [13–15]. For very small populations genetic drift is generally considered an important factor [13], as the theory successfully predicts the outcomes of neutral evolution experiments [9, 16].

In most theoretical analyses, constant (or effectively constant) population sizes are assumed, and the role of population growth is neglected. Bacterial populations, however, often undergo rapid growth—especially when they are small. For example, as few as 10 individuals of some highly virulent pathogens (e.g. enterohemorrhagic Escherichia coli or Shigella dysenteriae) suffice to initiate a deadly infection in a human host [17, 18]. Another case of small, growing populations are water-borne bacteria that feed on phytoplankton products. Due to nutrient limitation in open water, these bacteria typically live in small populations in close proximity to the planktonic organism [19]. During spring blooms, the phytoplankton releases more organic material, boosting the bacterial growth rate [19–21]. In nature, such small populations often form by adventitious dispersal from a larger reservoir population [22]. A typical example is the spreading of pathogens from host to host. This random “sampling” from a reservoir yields small populations whose genetic compositions differ from that of the reservoir (a phenomenon known as the founder effect [23]). Recent studies also showed that the combination of population growth and stochastic fluctuations can have a major impact on the evolution [5–7, 24] and genetics [25] of small populations.

To probe how population growth shapes genetic diversity, we used a well-characterized microbial model system, namely the soil bacterium Pseudomonas putida KT2440 [3, 29, 27]. The wild-type strain KT2440 produces pyoverdine, an iron-scavenging molecule that supports growth when iron becomes scarce in the environment. Here we consider co-cultures of two genetically distinct strains: the wild-type, pyoverdine-producing strain KT2440 (strain A) and the mutant non-producer strain 3E2 (strain B). We set up conditions of non-selective competition between these strains by using an iron-replete medium (casamino acids, supplemented with 200 μM FeCl 3 ). In this medium, production of pyoverdine is effectively repressed [27], such that both strains have the same growth rate and neither has an advantage (see S2 Table). Producer (KT2440 wild type) and non-producer (3E2) strains were first mixed and diluted to yield Poisson dilution conditions. Then we initiated a large number of subpopulations from this reservoir by pipetting aliquots of the cell suspension into the wells of a 96-well plate, thereby generating a large ensemble of subpopulations with a random distribution of initial cell number N 0 and producer fraction x 0 (Fig 1). Use of shaken liquid cultures ensured homogeneous well-mixed conditions for all cells in the same well (access to nutrients, oxygen, etc.), and exponential growth was observed in all cases (see S2 Fig).

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larger image TIFF original image Download: Fig 1. Schematic depiction of urn sampling and growth. (a) Schematic illustration of the random initial conditions. An infinite reservoir contains a diluted mixture of bacteria, a fraction of which are of strain A. We draw small volumes of liquid from the reservoir containing small, random numbers of individuals, which conform to a Poisson distribution with mean (determined by the dilution of the reservoir population). A certain fraction of this initial population is of strain A. The mean value of this fraction is equal to . We use these individuals to initiate populations in the wells of a microtiter plate, so that each population starts with a random size N 0 and a random fraction of A-individuals x 0 . (b) Illustration of the Pólya urn model. If a bacterial population is represented as an urn, each individual as a marble and each bacterial strain as a color, this urn model captures the essentials of bacterial reproduction in our populations. At each iteration, a marble is drawn at random and returned to the urn, together with another one of the same color. The probability of extracting a marble of either color is determined solely by its relative abundance, making the process non-selective (since no strain has inherent advantages, see S2 Table). The rate of growth in population size can be rendered exponential (see S2 Fig) by letting the waiting time between successive iterations be exponentially distributed (also known as Poissonization). https://doi.org/10.1371/journal.pone.0134300.g001

This experimental setting is well described within the mathematical framework of a Pólya urn model. Consider each bacterium in the population as a marble in an urn, and its genotype as the color of the marble (e.g. red for strain A, and blue for strain B). Population growth results from single reproduction events in which an individual randomly divides. This is mathematically equivalent to a stochastic event in which a marble is chosen at random from the urn and put back, together with another one of the same color. This random process, introduced by Eggenberger and Pólya [4], exhibits several important properties [28–31]. It is self-reinforcing: each time a marble is extracted, another one of the same color is added, increasing the likelihood of extracting a marble of that color again. In the context of bacterial populations, this means that every birth event for one strain makes it more likely that further birth events of that same strain will occur in the future. Note, however, that fixation, i.e., complete loss of one type of marble from the population, cannot occur, simply because in the Pólya urn model marbles are neither removed nor do they change their color. This fully reflects the experimental conditions: During exponential growth, rates of cell death are negligible, and within the observation time mutations will be extremely rare, given the population sizes considered. The bacteria in each well reproduce randomly at a per-capita (average) rate μ. To translate this to the urn model, drawing of a marble is assumed to be a stochastic Poisson process, with a “per-marble” rate μ (a procedure known as Poissonization or embedding [32, 33]). Mathematically, the growth process in then described by a Master Equation: The time evolution for the probability P(N A , t) of finding N A individuals of strain A at time t reads (1) where we have set the growth rate to μ = 1 in order to fix the time scale (for an introduction to the mathematical concepts see, e.g., [34]); the corresponding Master equation for individuals of strain B is of identical form. To study the composition of the populations, we use the more convenient quantities N = N A + N B (total size) and x = N A /N (fraction of individuals of strain A).

To start the experiment, we inoculated the wells of 96-well-plates by drawing small volumes of diluted liquid bacterial culture from a large reservoir (Fig 1(a)). Each volume contains a random number of bacteria whose mean value is controlled by the dilution of the reservoir. The fraction of bacteria of strain A (wild type) in that volume is also random, with its mean value given by the fraction of strain A in the reservoir. In the mathematical formulation, this setup corresponds to stochastic initial conditions for the Pólya urn model: the initial population size N 0 for each well is given by a Poisson distribution with mean , and each individual is assigned to strain A or B with probability and ), respectively. This procedure is also equivalent to treating the initial numbers of A- and B-individuals as independent, Poisson-distributed random variables with mean values and ), respectively [6].

Fig 2 shows a time series of the histogram for the composition x of all subpopulations considered, as obtained from a stochastic simulation of the Master Eq (1) for a given random initial condition (with and ). Surprisingly, the distribution first broadens, but then quickly “freezes” to a steady state (see S1 Video). This is genuinely different from Kimura“s result for populations with constant size [1] (or similar results with effectively constant size [2]) where the balance between stochastic birth and death events leads to genetic drift, and thereby eventually to the extinction of one of the two strains. In contrast, for a growing population, death events are negligible, and therefore there is no fixation of the population during growth. Instead, fixation arises as a direct consequence of the initial sampling process, as can be seen from the heights of the black bins in the histogram (at x = 0 and x = 1), which remain constant over time (Fig 2). During growth, the composition of each subpopulation, instead of drifting to fixation at either x = 0 or x = 1, reaches a stationary limit value x*, where it remains thereafter [35]. This limit value is random: starting several subpopulations (urns) from the exact same initial composition of strain A and B (blue and red marbles), each reaches a limit, but in general these limits differ from one another. Once all of the subpopulations in an ensemble reach their limit, the distribution of the population composition freezes to a steady state, which is equal to the probability distribution of x*. Similar random limit properties appear in other fields, with lock-in in economics as the best-known example [30].

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larger image TIFF original image Download: Fig 2. Time series for the simulated distribution of the population composition x. The distribution initially broadens, then freezes to a steady state (see S1 Video). The fraction of populations that have x = 0 or x = 1 (indicated by the black bins) remains constant during the time evolution, as expected for a Pólya urn process, and in contrast to expectations from genetic drift (see S1 Table). In each well the population follows a stochastic path and reaches a (random) limit composition, and the distribution freezes only when all populations reached their limit. The parameter values used in the simulation are and The inset shows the mean, standard deviation and skewness as a function of the number of generations, with symbols denoting numerical simulations, and the solid lines corresponding to the theoretical prediction of Eq (2) (and also those in S2 Text). Analytical and numerical values agree. The mean ⟨x⟩ remains constant throughout the evolution, as expected for a non-selective process; standard deviation and skewness saturate to limit values, confirming the freezing of the distribution. https://doi.org/10.1371/journal.pone.0134300.g002

The inset in Fig 2 shows approximate solutions for the time evolution of mean, standard deviation, and skewness of the composition x, which we obtained by analytically solving the Master Eq (1) (see S2 Text). The analytical results agree well with their numerical counterparts. In particular, the mean value remains constant over time, as it must for a non-selective process. For the time evolution of the variance, which is a measure for the spread of a distribution, we obtain to leading order in population size (2) The broadening and freezing of the distribution is reflected in the exponential decay term of the variance. Note that the skewness increases as well, because growth is self-reinforcing (see inset in Fig 2). To further test the validity of the stochastic simulations, we also calculated the limit values of the average and variance after extended periods of evolution exactly, and found that they match the numerical solutions of the Master Equation perfectly (see S1 Text).

We tested these theoretical predictions using P. putida as a bacterial model system. We mixed the wild-type and mutant strains in order to obtain different initial fractions . The degree of dilution of the mixture determines the average initial cell number , with which we inoculated 120 wells per experiment (96-well plate format). In order to compare the experimental data with our model, we set up simulations that matched the experimental configuration by initializing and with the same values as measured in the experiments. We simulated the time evolution of about 104 populations, grouped in “virtual plates” of 120 wells each. Every virtual plate produced a histogram like the one we obtained from experiments. We then generated an average histogram of the virtual plates and used its values to compute the binomial confidence intervals [36] for the count in each bin, and compared those with the experimental distribution.

Fig 3(a) shows a representative experimental histogram of the initial population sizes N 0 for strong dilution with . It is well approximated by a Poisson distribution, and agrees with the simulation results within statistical errors (blue line and shaded gray areas in Fig 3(a)). Fig 3(b) shows the probability distribution of the corresponding initial compositions x 0 of the populations, where again theoretical and experimental values agree well within statistical error. Note also that in every well the composition x 0 must be a simple fraction; this means that only a few numerical values are possible for small initial population sizes N 0 . This small-number effect explains why the distribution of x 0 in Fig 3(b) is so ragged. The distribution becomes much smoother for larger initial population sizes (see S3 Fig). Taken together, these results for the distribution of initial population size and composition confirm that the inoculation of the individual wells is a stochastic sampling process with Poissonian statistics.

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larger image TIFF original image Download: Fig 3. Initial distributions for population size N 0 and composition x 0 (parameter values Initial distributions for population size Nand composition x(parameter values). The experimental distributions (bars) for N 0 (panel (a)) and x 0 (panel (b)) are measured from 120-well ensembles. The average and calculated from the measured values determine the parameters for the simulated distributions. The theoretical average distribution (solid blue line) is the average of the same distributions generated for 84 sets of 120 wells. Using that average we calculate the Wilson binomial confidence intervals (gray areas) for 68% (between dashed lines), 95% (between dotted lines) and 99.73% confidence. The measured and simulated distributions agree well within statistical error, confirming our assumption that individuals of strain A and B in the experiments start Poisson-distributed with mean and , respectively. The ragged distribution of x 0 derives from a small-number effect, and disappears at larger values of N 0 (see main text, and also S3 Fig). https://doi.org/10.1371/journal.pone.0134300.g003

Next, we were interested in how the composition of the bacterial population would evolve under non-selective (neutral) growth conditions. To this end we let the 120 populations grow for an 11-hour period, during which they remained in exponential growth phase (see S2 Fig). Then we measured the population size N(t) in each well by counting colony-forming units, and x(t) by counting the pyoverdine-producing colonies (see Materials and Methods). Fig 4 shows the final outcome for four different initial conditions, i.e. combinations of the initial average population size and composition . We first wanted to know what determines the number of wells that contain only individuals of either strain A or strain B, i.e. that are fixated. To this end we compared the experimentally observed values with the corresponding predictions from the numerical simulations of the Pólya urn model (Fig 4). Since both results agree within statistical error, we conclude that fixation of a population is a consequence of the initial sampling process and is not due to fixation during population growth (see also S1 Table). This is especially obvious for small average initial population size or compositions close to x = 0 or x = 1, where a large fraction of the wells contains cells of strain A or B only (Fig 4(a) and 4(d)). Next we wished to learn how the final distribution of the population composition (i.e. the random limits, x*) depends on the initial average composition . For , we observed both by experiment and theoretically that the initial distribution significantly broadened (by a factor of ) but remained symmetrical (Fig 4(c) and S1 Fig). In contrast, starting from distributions with average values below or above 0.5 caused the final distribution to broaden and also become skewed towards smaller or larger values of x, respectively (Fig 4(b) and 4(d)). Moreover, we found quantitative agreement between experiment and numerical simulations within statistical errors in all analyzed parameter regimes (see blue lines and shaded areas in Fig 4): most experimental histograms fall within the first confidence interval of the prediction (darkest gray areas, between dashed lines), and almost all fall within the 99.73% confidence interval.

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larger image TIFF original image Download: Fig 4. Steady-state distributions of population composition x for different initial conditions. The experimental distribution (bars) is the result of growth on 120 independent wells. We use the measured average x 0 and N 0 from the experiments to initialize the simulations of several 120-well ensembles. After growth, we compute the histogram for each of these ensembles and obtain the average theoretical distribution (blue line). Using the values from this distribution, we compute the three confidence intervals (shaded gray areas) for each bin for 68% (between dashed lines), 95% (between dotted lines) and 99.73% confidence. The two sets of data match: most experimental data agree with the first prediction confidence region, practically all with the second one. The limit distributions are clearly different from the initial ones (see S1 Fig). The importance of growth in changing the distributions depends on the initial size N 0 (see main text, and S1 Fig). Parameter values: , (panel (a)); , (panel (b)); , (panel (c)); , (panel (d)). https://doi.org/10.1371/journal.pone.0134300.g004

Taken together, our combined theoretical and experimental analysis gives a coherent picture of evolution during non-selective (exponential) growth. We have shown, experimentally and by analogy with the Pólya urn model, that for each well-mixed population the composition of the population reaches a random stationary limit, and, unlike populations with constant size, generally does not fixate. For a large ensemble of populations, this implies that the probability distribution for the population composition converges to limit distributions (Figs 2 and 4), which are nothing like Kimura“s result for constant-sized populations. Our result is also quite different from that obtained in range expansion experiments [37] or other settings featuring population growth without death on two-dimensional substrates. There, monoclonal sectoring patterns arise as a consequence of random genetic drift, which drives population differentiation along the expanding fronts of bacterial colonies, unlike our well-mixed populations that freeze to coexistence.

Our study also shows that, in a growing population with stochastic initial conditions, demographic noise has two possible sources: the initial sampling process by which subpopulations are formed, and the subsequent growth process. The initial average population size sets their relative weight (see S3 Text and S1 Fig). For very small , of the order of one or two individuals, the formation process already determines the final composition distribution: most populations start off fixated, many with just a single founder individual, and the composition of each well remains the same during growth. For very large , of the order of a few hundreds, the sampling process is again central: the composition distribution changes very little before freezing, and growth generates only a very limited amount of variation. In these two limiting regimes, neglecting stochastic effects during growth leaves the evolutionary outcome practically unchanged. In contrast, for small founder colonies such as those typically found during population bottlenecks [18, 19, 38] ( ), population growth is responsible for the major part of the variation observed in the final distribution.

Moreover, our results reveal that a growing population reaches a random limit composition much faster than genetic drift leads to fixation in populations of constant size. Typical fixation times for genetic drift increase logarithmically with the population size [11], while the time scale for freezing is independent of population size. This has important consequences for the role of stochastic effects when a population passes from exponential growth phase to stationary phase, in which growth rate and death rate are equal. Then, the composition of the population shows both freezing and fixation, albeit at quite distinct times because the relevant time scales differ markedly. During growth the composition distribution quickly freezes, as described above. Once the population reaches its stationary size, it slowly drifts to fixation, following Kimura-like dynamics.

We also believe that our results have a broad range of applications since growing populations are ubiquitous in nature. For example, experimental studies of P. aeruginosa [22, 39] have shown that typical life cycles pass through different steps with regularly occurring dispersal events being followed by the formation of new colonies. As initial colony sizes are typically small, such dispersal events coincide with population bottlenecks and subsequent exponential growth. During these phases of the life cycle, population dynamics is often selectively neutral and hence falls within the framework of the present work. The degree of diversity generated during these population bottlenecks has been shown to be crucial for some proposed mechanisms for the evolution of cooperation under selective pressure [5–7, 40–42]. Our analysis quantifies the ensuing degree of diversity and points to the relative importance of sampling versus growth for long-term behavior of the reservoirs. This may have important consequences for the degree of genetic diversity observed in natural populations with life-cycle structures [38].