Mathematical modelling generates predictions on the plasmid replication mode based on the empirical plasmid allele dynamics

To formulate the theoretical expectation for the allele dynamics in the experiments, we adapted a previous mathematical model of vertical replicon inheritance in bacterial populations to match the experimental design of the evolution experiment (Fig 1C; refs. [27,29]). The genotype of a cell is defined by the number of novel and ancestral replicon copies at cell birth, with the total number of novel replicon copies ranging between 0 and the replicon copy number n_rc. Cells in the simulations of monoploid replicons (n_rc = 1), such as the chromosome in A. baylyi, harbor either the ancestral or the novel allele. Cells in the simulations of multicopy replicons (n_rc > 1), e.g., model plasmid pTAD-R, can be homozygous for the ancestral or for the novel allele, or heterozygous, carrying both allele variants in various proportions. We consider thus all n_rc+1 possible cell types. Our model includes two possibilities for the mode of replication: under the regular replication mode, all n_rc replicon copies are duplicated before a cell divides. Under the random replication mode, single copies are selected randomly for replication in a repetitive process until the cell carries 2 × n_rc copies (refs. [27,29]; see Methods for details). For either mode of replication, the replicon copies are distributed randomly into the two daughter cells at cell division (similarly to pTAD-R), where each cell inherits n_rc copies. Population growth between bottlenecks is modelled by deterministic logistic growth, where we only account for cell replication but not for cell death. At bottlenecks, cells are randomly sampled, introducing stochasticity (see detailed model description in the methods).

The replication mode of the pTAD-R backbone has not been examined experimentally so far. To determine which modelled mode of replication better matches the experimental system, we simulated the segregation dynamics of a novel allele carried by a polyploid replicon and compared the simulated dynamics to the experimental results. The simulation was performed for both replication modes under neutrality (s = 0) with a replicon having n_rc = 15, as model plasmid pTAD-R. In this simulation, we opted for the highest initial frequency, f₀ = 10⁻⁴, and the weak bottleneck, b = 10⁻², where stochastic effects other than segregational drift are weakest. The model results show that the frequency of heterozygous cells rapidly decreases at a constant rate, while the frequency of homozygotes first increases and then reaches an equilibrium around transfer 15 (Fig 1D). The comparison between the frequency of heterozygotes and homozygotes from simulations under the mode of regular replication and the experimental results shows a good qualitative agreement for the first 11 transfers (Fig 1D). From transfer 12 onwards, the model underestimates the frequency of heterozygous cells by up to ten-fold. The deviation between the model prediction of the homozygotes frequency and the experimental results increases from transfer 15 when the homozygotes frequency in the experiment decayed. Simulations using the random replication mode overestimate the decay of heterozygotes’ frequency as well as the increase in homozygotes’ frequency already during the very first transfers (Fig 1D). To test for possible reasons for the observed deviations, we performed further simulations under the regular replication mode, assuming a different plasmid copy number (n_rc between 10 and 30) or a slight negative fitness effect of 1–5% of the novel allele. Increasing the copy number in the simulations yields a better fit to the heterozygote frequencies (S1A Fig); nonetheless, the experimental quantification of pTAD-R copy number does not suggest a large deviation from n_rc = 15 [28]. Simulations with a negative selection parameter (approximately between s = -0.01 and s = -0.03) yield a better fit to the homozygote frequency observed in the experiment (S1B Fig). Indeed, while the experimental results suggested no significant fitness effect of the novel plasmid allele, the average relative fitness suggested a slight deviation from a neutral regime (S2 Fig). Note that random replication always entails faster segregation dynamics than regular replication [29]. Hence achieving a good fit between the simulation with random replication and the experimental results would require a substantial increase in the plasmid copy number beyond the experimental evidence for pTAD-R. Consequently, we opted to use the regular replication mode for further simulations of pTAD-R allele dynamics. The correspondence between the simulation results under the regular replication mode and the allele dynamics in the evolution experiment suggests that the replication of plasmid pTAD-R is not random and may follow a regular mode of replication.

Beneficial alleles on polyploid replicons are prone to loss due to segregational drift

To examine the effect of replicon polyploidy on the dynamics of beneficial alleles, we experimentally evolved the bacterial populations under conditions where the novel allele is beneficial, i.e., under selection for kanamycin resistance. In order to identify a comparable selection regime for both replicon types, we quantified the fitness effect of the nptII allele when encoded in the multicopy plasmid or single copy chromosome, employing increasing kanamycin concentrations (S5 and S6 Figs). During the evolution experiment, we applied a constant antibiotic pressure, which confers cells that are homozygous for the novel allele a fitness benefit of ca. 10% (i.e., selection coefficient s = 0.1185 ± 0.0415 SD for the novel resistant strain). The concentration of antibiotics employed to achieve a fitness benefit of ca. 10% in multicopy and single-copy replicon in novel homozygotes is similar. Therefore, we assume that the effect of gene dosage in our system is negligible; that is having more than one copy (1 to 15 copies) does not produce a higher fitness benefit to the bacteria at the concentrations employed. Note that the effect of selection is restricted to processes at the population level; antibiotic selection has no known effect on plasmid replication and segregation. We examined the effect of selection on the dynamics of the novel allele in combination with the effect of initial frequency and population bottleneck size. For each replicon type, we thus tested a total of 12 parameter combinations, each with 6 replicates (Figs ​(Figs2,2, S3 and S4).

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Fig 2

Plasmid and chromosomal allele dynamics under selective and non-selective conditions.

(A) The dynamics of a plasmid allele over approximately 200 generations. Points correspond to the proportion of kanamycin-resistant cells in each population and sampling time. Stacked bar plots present the state of the kanamycin-resistant populations at the end of the experiment. The fixation threshold is set at a frequency of 0.9. (B) Plasmid allele dynamics of representative populations from experimental conditions 2, 6 and 10 evolved under selective conditions and weak bottlenecks. The white arrow marks the point at which the frequencies of heterozygous and novel homozygous cells are equal. The remaining experimental populations are shown in S3 and S4 Figs). (C) The dynamics of a chromosomal allele over approximately 200 generations.

We first focus on the results for the plasmid-encoded allele. As expected, the novel allele is maintained for longer and lost less often under selection than under neutral conditions. With high initial frequencies (f₀ = 10⁻⁴), the fraction of kanamycin-resistant cells increased in the population over time, regardless of the population bottleneck (Fig 2A, plots 1,3). The frequency of cells that carry the novel allele at the end of the experiment is close to the fixation threshold (that was set to 90%) in most replicates. In contrast, at low or moderate initial frequencies, the novel plasmid allele was quickly lost in most populations (Fig 2A). In populations that evolved under weak bottlenecks, the novel plasmid allele is lost in 1 out of 6 replicates for moderate initial frequencies and in 4 out of 6 replicates for low initial frequencies (Fig 2A, plots 5,9). In some replicate populations where the allele is maintained, we observed a slow decrease in the fraction of kanamycin-resistant cells from the 20^th transfer on. Under the strong bottleneck, none of the replicate populations maintained the novel allele for moderate or low initial frequencies (Fig 2A, plots 7,11).

The combination of selection and initial frequency has an effect also on the proportion of of homozygous and heterozygous cells within the resistant sub-population. For high initial frequency (f₀ = 10⁻⁴) under selective conditions, the frequency of cells that are homozygous for the novel allele increases rapidly and reaches a similar frequency to that of the heterozygotes within ca. 5–10 transfers (Fig 2B, plots 2.2 and 2.6). This initial rise in frequency is faster in comparison to that observed under non-selective conditions (ca. 10–15 transfers; Fig 1B). Under selection, both resistant genotypes, heterozygotes and homozygotes, are maintained in the population for a longer period of time and the frequency of heterozygotes decreases more slowly than under non-selective conditions. Nonetheless, heterozygotes are eventually lost from the population in several replicates (from ca. 25th onwards) (Figs ​(Figs2B,2B, S3 and S4). With medium and low initial frequencies, the homozygotes’ frequency increases more slowly, such that similar frequencies of homozygotes and heterozygotes are observed later in the experiment (Fig 2B). In summary, under selective conditions, both resistant genotypes are maintained for a longer time, where the relative frequencies of homozygotes and heterozygotes change as a consequence of segregational drift.

To further examine the effect of polyploidy on the dynamics of novel alleles, we compared the dynamics observed for the plasmid-encoded allele to those observed for the chromosome-encoded allele. When the allele is encoded on the plasmid, the fraction of kanamycin-resistant cells increases more slowly and the time to fixation is longer in comparison to the chromosome-encoded allele (Fig 2C). The slower increase of the frequency of kanamycin-resistant cells in the experiments with the plasmid-encoded allele is best observed at mid-experiment (ca. transfer 15) for the high initial frequency (f₀ = 10⁻⁴), where the frequency of kanamycin-resistant cells is 100 times lower in comparison to the chromosome experiment. At the end of the experiment, the kanamycin resistance phenotype was fixed in a single population within the plasmid experiment, in contrast to the chromosome experiment where the resistance phenotype was fixed in the majority of populations (Fig 2A and 2C, plots 1,3). At lower initial frequencies (f₀ = 10⁻⁶, f₀ = 10⁻⁷), the differences between the frequency of kanamycin-resistant cells observed in the plasmid and chromosome experiments are even more pronounced, with up to ca. 1000-fold in favour of the chromosome populations (Fig 2A and 2C, plot 5). The most striking difference between plasmid and chromosome experiments is observed in populations that evolved under a strong bottleneck (b = 10⁻³). In those populations, the plasmid-encoded allele is lost in all replicate populations (Fig 2A and 2C, plots 7,11), in contrast to the chromosome experiment where the allele is maintained. To conclude, the opposing effects of segregational drift and selection lead to longer fixation time and loss of plasmid alleles in comparison to chromosomal alleles. Our results demonstrate that segregational drift has a pivotal role in the evolution of polyploid replicons.

An alternative explanation for the observed loss of the novel beneficial plasmid allele could be the adaptation of the originally kanamycin-sensitive cells to the antibiotic environment, such that the novel allele is no longer beneficial. To rule out this alternative, we performed a competition experiment where the ancestral kanamycin-resistant populations competed against the evolved kanamycin-sensitive populations (S7 Fig). Our results show that the evolved kanamycin-sensitive populations still had a lower fitness under selective conditions compared to the ancestral kanamycin-resistant populations. Consequently, we conclude that the dynamics of the novel plasmid allele observed during the evolution experiment are the result of plasmid allele segregation rather than adaptation of the kanamycin-sensitive population to kanamycin.

The negative effect of segregational drift on the fixation of the beneficial allele is predicted by model simulations

The experimental results in the previous sections are in line with previous theoretical predictions on the effects of replicon polyploidy on the dynamics of beneficial alleles [27]. For a direct comparison of the experimental results to theoretical predictions, we performed simulations with the model that is adapted to the experimental set-up. In our simulations, we assume that cells carrying only the ancestral allele replicate at a rate 1–s, and cells carrying at least one copy of the novel allele replicate at a rate 1. The model assumes no effect of gene dosage.

To obtain a selection parameter that optimally reflects the allele dynamics over the entire duration of the experiment, we first estimated the parameter s from the mean frequencies of the kanamycin-resistant subpopulation in the evolution experiment for large initial frequencies (f₀ = 10⁻⁴) and weak bottlenecks (b = 10⁻²). The s parameters yielding an optimal fit between the simulated dynamics of the novel allele and the corresponding dynamics were selected for further simulations (S8 Fig). Those deviated from the experimental assessment of the fitness benefit of the novel allele: s = 0.087 ± 0.016 for the polyploid replicon (n_rc = 15) and s = 0.136 ± 0.010 for the monoploid replicon (n_rc = 1) (see S8 Fig).

To examine how well our model captures the experimental results, we first performed simulations that correspond to the experimental conditions (Fig 3). The dynamics of the novel phenotype (kanamycin resistance) were overall in agreement with the results of the evolution experiment for both replicon types, albeit the frequency of the novel phenotype is slightly overestimated by the model given low or moderate initial frequencies (f₀ = 10⁻⁶ or 10⁻⁷) or strong bottlenecks, b = 10⁻³. The simulation results for the polyploid replicon show that the rise in frequency of heterozygotes in the experiment is well captured by the model simulations for high and moderate initial frequencies (f₀ = 10⁻⁴ and 10⁻⁶) and bottleneck size combinations. Nonetheless, the model underestimates the frequency of heterozygotes observed in the evolution experiment, which is consistent with the overestimation of the loss rate of heterozygotes under neutral conditions. The frequency of heterozygotes in the simulation starts decaying at nearly the same transfer as in the experiment, yet heterozygotes in the evolution experiment reach frequencies that are 10- to 100-fold higher in comparison to the simulations. Note that a comparison of the simulated allele dynamics to the plasmid experiment for the low initial frequency (f₀ = 10⁻⁷) is not possible, as most replicates in those experiments went extinct. Using the model simulation for the low initial frequency thus supplies a theoretical expectation for the beneficial allele dynamics in rare cases where the novel allele may persist in the population. The simulation results for moderate and low initial frequencies (f₀ = 10⁻⁶ and 10⁻⁷) predict a higher variability in the kanamycin-resistant cell frequencies per transfer for the polyploid replicon in comparison to the monoploid replicon (Fig 3A, plots 5–12). This prediction conforms to the results of the evolution experiments (S9 Fig). To further test the robustness of the comparison between theory and simulations regarding the choice of the parameter s in the model, we further performed computer simulations with s = 0.1. The resulting dynamics were qualitatively in agreement with our previous results as well (S10 Fig). We further compared the survival probability of the novel allele over time with the experimental results for the fraction of populations in which the allele is still present after a given transfer (Fig 3B). The model showed a remarkable agreement with the experimental results, capturing the main trends and often even making quantitatively accurate predictions.

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Fig 3

Model simulations of beneficial allele dynamics for polyploid (n_rc = 15) and monoploid replicons.

Predicted frequencies of the novel phenotype, given has not (yet) gone extinct, are shown in (A) and the corresponding survival of the novel allele are shown in (B). (A) Cell frequencies of the novel phenotype (parallel to kanamycin-resistant cells) (black) and heterozygous cells (green) at the end of each population transfer obtained from 100 simulations for a replicon with copy number n_rc = 15 (1,3,5,7,9,11, for the multicopy plasmid) and with n_rc = 1 (2,4,6,8,10,12, reflecting the chromosome). Lines show the mean of positive frequencies over the number of transfers, i.e., the average proportion of cells having the novel phenotype from all simulation trajectories, where the novel allele has not (yet) gone extinct (cf. (B) for the survival of the allele). The initial population at the start of the first transfer consists of ancestral cells and a small proportion, f, of cells with one novel replicon copy, where f₀ = 10⁻⁴ in (1–4), f₀ = 10⁻⁶ in (5–8), and f₀ = 10⁻⁷ in (8–12). The grey (dark grey) areas show the 99% (67%) confidence intervals of positive frequencies of the novel phenotype. The green lines show the mean frequency of heterozygous cells carrying both the ancestral and novel plasmid allele (conditioned on positive frequencies). Grey and green markers show the frequencies of kanamycin-resistant cells and heterozygous cells, respectively, of the corresponding experimental replicates. (B) Survival of the novel allele over number of transfers in model simulations (lines) and in the evolution experiment (markers) f₀ = 10⁻⁷. The survival of the novel allele initially present at high initial frequencies, f₀ = 10⁻⁴, are not shown since it is always 100%. The fraction of kanamycin-resistant cells of 100 simulation trajectories and 6 experimental replicates, respectively, is shown for each parameter combination. Simulation results were obtained by integrating Eq (1) with s = 0.087 for the polyploid replicon and s = 0.136 for the monoploid replicon until the population size has reached N_c(t) cells in each growth phase (transfer), where N_c(t) is taken from the mean experimental carrying capacity of each transfer t. A population bottleneck is applied after each growth phase, where a proportion of b = 10⁻² or b = 10⁻³ cells are randomly selected from a multinomial distribution defined by the genotype frequencies at the end of the growth phase.

Fitness experiments

The relative fitness (w) of the ancestral allele carrying strain versus the novel allele carrying strain was estimated by direct pairwise competition experiments, with 8 replicate populations for the plasmid replicon and 6 replicate populations for the chromosomal replicon. The determination of the selective coefficient given by the selection treatment was calculated via determination of the relative fitness of the ancestral allele carrying strain versus the novel allele carrying strain in direct pairwise competition experiments under selective conditions with 6 replicates populations for both plasmid and chromosomal replicons. The relative fitness (w) of the ancestral kanamycin-resistant populations at the beginning of the experiment vs the evolved kanamycin-sensitive populations at the end of the experiment was estimated by direct pairwise competition experiments with 8 replicate populations per experimental condition tested (conditions 8 and 12). All competition experiments were initiated with a 1:1 ratio of the competing strains, diluted 1:100 from overnight cultures, pre-conditioned to growth in liquid media for two days. The relative fitness is calculated by calculating the growth of each of the competitors after 24 hours of growth, the cultures are serially transferred for 5 days and the populations determined. Serial transfer was done in non-selective media for ancestral carrying vs novel allele carrying population competition and on selective media (kanamycin 0.375 μg/ml) for selective coefficient determination and kanamycin-resistant vs non-resistant competition. The two competitors are distinguished by plating on non-selective (LB supplemented with IPTG 1mM) and selective media (LB supplemented with kanamycin 10 μg/ml for novel allele carrying strains (chromosome or plasmid) or M9 for ancestral allele carrying chromosomal strains).

Natural transformation of Acinetobacter baylyi

The preparation of competent cells of A. baylyi carrying pTAD-R was prepared as described previously in ref. [45]. Briefly, the cells were grown at 30°C overnight with shaking in 2 ml of LB medium and used for the inoculation (1:100) of fresh cultures. These cultures were grown until early stationary phase (approx. 1×10⁹ cells per ml), cooled down and then stored as concentrated stocks (1×10¹⁰ cell per ml) at -80°C.

For each transformation performed in the study, the frozen competent cells were thawed on ice and diluted to a cell density of 2.5×10⁸ cells per ml. The donor DNA was a PCR fragment (for a detailed description see ref. [28]) containing the non-disrupted nptII⁺ as the single segment of sequence identity to the recipient plasmid. After 90 minutes of incubation at 30°C with shaking, diluted aliquots of the culture were plated on LB medium to estimate the titer of the population and transformants were scored on selective media. Transformation frequencies were calculated as transformants per recipient.

Natural transformation in A. baylyi follows one-hit kinetics [46], where the number of created transformants is proportional to the concentration of donor DNA added in the transformation process. The uptake probability of DNA molecules per host cell does not change with increasing concentration of DNA. Therefore a single donor DNA molecule recombines with the host DNA in the vast majority of cells (97% [28]).

Evolution experiments

The plasmid allele dynamics evolution experiment was conducted with ancestral plasmid carrying A. baylyi populations under two selection regimes, three novel initial allele frequencies and two population bottleneck sizes. The ancestral plasmid present was pTAD-R for all populations (n = 72). On the onset of the experiment, A. baylyi strain BD413 containing the ancestral plasmid was naturally transformed with the donor DNA as explained in the previous section following the previously described protocol [28]. Donor DNA was added at specific concentrations to generate diverse transformants frequencies. Three different initial allele frequencies were created: 10⁻⁴ (700ng/ml donor DNA), 10⁻⁶ (7ng/ml donor DNA) and 10⁻⁷ (0.7ng/ml donor DNA). After the transformation was complete (i.e., after 90 minutes of incubation), a fraction of the cells was plated to determine the total cell population and the transformation frequency. The viable titer of the cultures was obtained from the cells plated on LB media (10⁻⁴, 10⁻⁵ and 10⁻⁶ dilutions), whereas the transformation efficiency was calculated by plating on LB + kanamycin (5 μg/ml) (1, 10⁻¹ and 10⁻² dilutions). Cells that had restored the nptII gene on one of copies of the recipient plasmid survived the kanamycin treatment. After overnight growth cultures were diluted into fresh medium with 1:100 (weak bottleneck) or 1:1000 (strong bottleneck) dilution factor. The bacterial populations were subjected to two selective regimes, non-selective conditions and selective conditions (kanamycin 0.375 μg/ml). The cultures were incubated in 96-deep-well plates with a total volume of 1 ml at 30°C with constant shaking. The transfer regime was then maintained for 30 transfers, which corresponds to approx. 200–300 generations (~6–7 generations per transfer in liquid medium for the weak bottleneck and ~9–10 generations for the strong bottleneck). Every day the population sizes were estimated, the total population size was estimated from the plating on LB media and the proportion of hosts in the population was measured by plating on selective media, LB supplemented with kanamycin. The distinction between heterozygotes and homozygotes for the new allele (nptII) comes from the gene of the reporter construct. For our model plasmid, the fluorescence that confers the GFP protein enables us to distinguish them.

The chromosomal allele experiment was performed by employing a mixture of two A. baylyi strains, BD4 (tryptophan prototroph) carrying a tetA gene which is the ancestral allele in this setting and BD413 (tryptophan auxotroph) carrying a nptII gene which is the novel allele. The strains were then mixed in three ratios comparable to the initial novel allele frequencies of the plasmid experiment, BD4/BD413 at 10⁻⁴, 10⁻⁶ and 10⁻⁷ proportions. A fraction of the cell was plated to determine the initial host proportion and the cell titers were calculated as previously described employing M9 minimal media and LB supplemented with kanamycin (5 μg/ml) as selective media to distinguish both strains. The experimental set up was identical as the one employed for the plasmid allele dynamics experiment, with serially passaged populations employing a 1:100 or 1:1000 dilution factor under non-selective and selective conditions (n = 72). The bacterial populations were subjected to two selective regimes, non-selective conditions and selective conditions (kanamycin 0.4 μg/ml). The cultures were transferred for 30 transfers and population size estimation was done for each day.

Statistical analysis

All statistical tests and data analysis were performed in RStudio version 1.2.1327. Comparisons between groups (e.g., fitness calculations) were performed with two sided Wilcoxon signed-rank test [47] by the “wilcox.test” function.

Simulation

We simulate the population dynamics of the serial dilution experiment using a random segregation model for multicopy replicons [27,29], which we extended for periodic bottlenecks. The model comprises competitive growth of bacteria, periodic bottlenecks, and plasmid segregation. In our model, cells with at least one copy (i≥1) of the novel (resistance-conferring) plasmid variant have an intrinsic cell division rate r_i = 1, where i denotes the cell type. Bacteria without the novel plasmid variant have an intrinsic cell division rate r₀ = 1−s, where s>0 reflecting an antibiotic environment or s = 0 for a neutral environment. We ignore cell death. At the beginning of the simulation, the population consists of N⁽⁰⁾ = 10⁷ cells with a (resistant) sub-population of proportion f₀ that carries one (out of n_rc) replicon copy of the novel variant. All cells have a fixed copy number of the plasmid n_rc at cell birth. We describe two modes of plasmid replication during cell division, regular and random replication. For regular replication, all plasmid copies are duplicated in a parental cell before it divides. For random replication, plasmid replication follows a successive manner. First, one plasmid is randomly chosen from the plasmid pool for replication and the two replicates are added to the pool of plasmids (see also [33]). This procedure is repeated n^rc-1 more times such that a cell–as for regular replication–carries 2n_rc plasmid copies before cell division. Finally, plasmid copies are distributed to the bacterial daughter cells at random, but in equal numbers to comply with the fixed plasmid copy number n_rc [cf. 27,29]. The influence of the assumption of fixed copy numbers on rescue probabilities (and thus indirectly establishment probabilities) on multicopy plasmids was discussed previously and shown to be small, especially for dominant mutations [27]. Bacterial populations grow logistically up to a carrying capacity ${\widehat{N}}^{\left(B\right)}$ before the transfer (bottleneck) B occurs, which we determine separately for each individual transfer and condition as the mean over the six experimental replicates. We model population growth between bottlenecks deterministically, which is much faster to simulate than a fully stochastic model. This model implementation is based on the assumption that bottlenecks (see below) are the dominating source of stochasticity. We, thus, have a system of n_rc+1 differential equations that describe the temporal change of the cell numbers N_i for the individual cell types i = 0,…,n_rc (cells carrying i novel plasmid copies) between two bottlenecks, B-1 and B.

Cells of any type j divide at a rate $r_j(1-\frac{{\sum }_{l=0}^{n_{rc}}{N}_l}{{\widehat{N}}^{\left(B\right)}})$. We thus obtain

where m_j→i denotes the expected number of i-type cells produced at division of an j-type cell. Kronecker’s delta (δ_ij = 1 for i = j and 0 otherwise) takes into account that the mother cell is replaced by two daughter cells at cell division.

For the expressions for m_j→i for regular and random replication, we refer to Appendix A in [29].

We define the end of the growth phase by the time when the time derivative of the various cell numbers, N_i, becomes very low, $\dot{{|N}_i|}\le 0.01$ for all cell types i, i.e., the change in the cell number of any cell type is at maximum 1% of the per-capita growth rate of wild-type cells. At this time, the carrying capacity is almost reached as the change in the total cell number ${\sum }_{i=0}^{n_{\mathrm{r}\mathrm{c}}}\dot{N_i}<{0.01n}_{\mathrm{r}\mathrm{c}}$ becomes very low. With this implementation, the number of cell divisions between bottlenecks in the simulations is expected to be similar to that in the experiments. At the end of the growth phase, populations undergo a bottleneck of factor b, which we model stochastically by drawing a proportion b of the population at carrying capacity (rounded to an integer number) using a multinomial distribution generated by the proportions of all cell types i in the population.

We thank Daniela Kluger and Ian Dewan for their assistance in the experimental work. The authors thank Yiqing Wang, Shreya Vichare and Ishan Bhatt for critical comments on the manuscript, the entire Genomic Microbiology group and Stochastic Evolutionary Dynamics group for their help and discussions.