Two cell lines derived from a single mouse mammary carcinoma – 168 and 4T07 cells – have similar growth rates when cultured individually, yet the 4T07 clone displays a dominant phenotype when grown together, either in cell culture or in orthotopic allografts in vivo (Miller et al., 1988). Several hypotheses to account for this interesting behavior had been tested in the original work, but the precise mechanism behind these competitive interactions has so far not been identified.

We began by verifying that in our hands the lines maintain their competitive characteristics. To facilitate lineage tracing we first generated lines stably expressing GFP, the expression of which did not alter cell growth (Figure 1A). Next, we followed growth characteristics of 4T07 and 168FARN cells, the latter being a drug-resistant derivative of the original 168 clone (Aslakson et al., 1991), in a continuous culture for 3 weeks. The cells were seeded as 1:1 mix at a density that allowed them to reach confluence within 3–4 days, at which point they were harvested and re-seeded in a new well at the original density. Remaining cells were analyzed by flow cytometry to determine the proportion of GFP expressing clones in the expanding population.

The homotypic co-culture (same line with and without GFP) confirmed that GFP has no impact on cellular proliferation (Figures 1B, 2B). In contrast, heterotypic co-culture conditions (two different lines, one expressing GFP) revealed the dominance of the 4T07 clone (Figures 1B, 2B).

These results confirm the originally described ecological interaction between the clones: 4T07 gradually dominates the culture while the 168FARN cells become scarce within 15–17 days. Importantly, the dominant phenotype is independent of the starting ratio between the two cell lines (Supplementary Figures 1A,B).

As originally discussed for the two clones under study (Miller et al., 1988), the expansion of a single clone in co-culture could be due to alterations in cell death or changes in the proliferation rates of either or both clones. We measured apoptosis in the loser 168FARN clone and found identical, very low levels of cell death under homotypic and heterotypic conditions (Supplementary Figure 2A). Next, we used time-lapse microscopy to assess the growth dynamics of both clones in continuous culture. The cells were seeded at a density that allowed reaching confluence in 4 days and were photographed every 45 min for the last 3 days. We measured the overall pixel intensity for each frame (Figure 3A) as a proxy for the growth rate of the fluorescently tagged cell line. This analysis revealed that under co-culture conditions, the growth rate of 168FARN decreased, whereas that of 4T07 increased relative to mono-cultures. To test whether increased net growth of the winner population is due to the alteration of proliferation, we estimated the proportion of cells in the S phase of the cell cycle by performing pulse-chase EdU staining. The results presented in Supplementary Figure 2B confirmed that heterotypic co-culture gave rise to significant decrease in cells actively replicating DNA for the loser clone and a significant increase in the winner clone. Overall, these results suggest that the dominant phenotype displayed by the winner cells in co-culture can be explained by changes in proliferation that operate in opposing directions on the winner and the loser cells.

To gain further insight into the ecological interactions between the winner and loser cell types we turned to mathematical modeling. Examination of the growth curves revealed two distinct phases of evolutionary dynamics (Figures 3A,B). In phase 1, from 0 to 45 h, the two cell types grew exponentially in both homotypic and heterotypic cultures, and the growth rate of 168 was higher than that of 4T07. This first phase can be regarded as a transition period before the cells start altering and responding to their new environment. By contrast in phase 2, from 45 to 72 h, the growth curves were strongly affected by interactions within and between the two cell types, and 4T07 grew faster than 168. To enable us to determine the mode of the ecological dynamics in each phase, we opted for a parsimonious, piecewise mathematical model. Specifically, we assumed a model with exponential growth in phase 1 and a transition to density-dependent competitive Lotka–Volterra-type dynamics in phase 2.

By fitting our model to the homotypic growth curves, we inferred the values of the phase 1 and phase 2 growth rates and the within-type interaction parameters (see section “Materials and Methods”). To infer the between-type interactions, we used additional data from 72-h competition assays, covering a wide range of initial ratios of the two cell types. Although this latter data set comprises only the initial and final proportions (at the beginning of phase 1 and the end of phase 2), we were able to infer the proportions at the beginning of phase 2 by adjusting for the exponential growth of both types during phase 1. We then used these inferred proportions and our previously inferred parameter values to estimate the remaining interaction parameters (see section “Materials and Methods”). The resulting model gives a good fit to the competition assay data (Figure 2A, first column) and is consistent with heterotypic time-lapse data not used for parameter inference (Figure 3 and Supplementary Figure 6).

The inferred parameter values (Table 1) imply that during phase 2, 4T07 has a large negative effect on both itself and on 168, consistent with 4T07 producing a harmful diffusible factor. The negative effect of 168 on itself is only about half as large, and 168 has approximately zero net effect on the growth of 4T07. This suggests that ubiquitous negative effects of 168 on 4T07 (e.g., likely due to waste products and competition for resources) are offset by positive effects, such as due to a beneficial diffusible factor. Also, during phase 2, the intrinsic growth rate of 168 (that is, the inferred growth rate before accounting for cell–cell interactions) is approximately 30% lower than that of 4T07, consistent with the conventional hypothesis that producing beneficial factors is costly. This disadvantage is offset by 168 having an approximately 30% higher carrying capacity (defined as the upper limit of the homotypic population size). Over phase 2, or any longer period that includes phase 2, the inferred net growth rate of 4T07 (that is, the growth rate after accounting for cell–cell interactions) is invariably higher than that of 168, which means 4T07 will come to dominate numerically, no matter their initial frequency.

Since we also conducted 96-h competition assays, we were able to infer the population dynamics during a third phase (72–96 h). For every initial ratio of the two cell types, the growth rate difference (also known as the gain function) was on average lower in the 96-h than in 72-h competition assays (Supplementary Figure 5). Moreover, this difference did not depend on the initial ratio, which implies it was not caused by a change in interaction parameters. A parsimonious way to account for this effect is to assume a reduction in 4T07’s intrinsic growth rate during phase 3, as would be expected to result from starvation and/or the build-up of toxic waste products. Making this adjustment to our model indeed produces a better fit to the competition assay data (Figure 2A, middle column, Figures 2B,C). The predicted dynamics are shown in Figures 3C,D.

Finally, having inferred all the evolutionary parameter values, we calculated net growth rates of the two cell types, averaged over different time periods. Over any period that includes phase 2, our model predicts that the net growth rate of both cell types will decrease non-linearly with increasing initial 4T07 frequency (pink and blue curves in Figure 2A). However, the net growth rate of 4T07 decreases faster than that of 168, which is why the gain function (gray curve in Figure 2A) also decreases. In phase 3, if the initial proportion of 4T07 is high (above 70%), then 168 has a higher net growth rate than 4T07, but in this case both of the inferred net growth rates are negative. Overall, the interactions are effectively equivalent to those of a parasite and its host, such that the “loser” 168 suffers from the presence of the “winner” 4T07, while also enhancing the winner’s fitness.

To identify the molecular mechanisms at the basis of the altered growth of winners and losers when in co-culture, we first focused on the increase in proliferation rate of 4T07 cells. Heterotypic culture experiments performed at low cell density suggested that the dominant effect did not require extensive cell–cell contacts (Supplementary Figure 3). We reasoned that a soluble factor secreted by 168FARN could induce a proliferation boost in 4T07. To test this hypothesis, we collected conditioned media from each line cultured for 3 days and used each medium separately to grow 4T07 for an additional 24 h. As controls, we either left the 4T07 medium after the 3 days of conditioning or replaced it with fresh medium. The results shown in Figure 2A confirm our hypothesis: the medium conditioned by 168FARN induced a significant increase in 4T07 proliferation. Importantly, this effect was not due to differences of medium exhaustion by the two cell lines, since the addition of fresh medium did not boost 4T07 proliferation.

Since our data strongly suggested that a soluble factor originating from 168FARN accounted for the increase in 4T07 proliferation, we next sought to define its molecular nature. First, we separated the 168FARN-conditioned medium into high and low MW fractions with a 3 KDa molecular cutoff column. The low MW fraction contains mainly metabolites while the high one is enriched in proteins. After complementing each fraction, respectively, with 10% serum or with DMEM to obtain full media conditioned with either low or high MW secretomes, we used them in a proliferation assay as in Figure 4A. The results (Figure 4B) of this series of experiments unambiguously identified the low MW fraction of the 168FARN-conditioned medium as the source of the pro-proliferative factor. To further explore its identity, we employed nuclear magnetic resonance spectroscopy to compare the composition of low MW fractions prepared from fresh medium and from the 168FARN- and 4T07-conditioned ones (Henke et al., 1996). Two major peaks specific for the conditioned media corresponded to a very strong signal for lactate secreted by 4T07 cells, and a significant increase in a peak identified as β-hydroxybutyrate (BHB) in the 168FARN-conditioned medium (Figure 5A). BHB is a ketone body mainly produced by the liver after long fasting periods and which is used by different tissues as a source of carbon to supplement the lack of glucose (Newman and Verdin, 2017). In addition, BHB is also produced by other cell types, such as adipocytes or cancer cells (Grabacka et al., 2016; Huang et al., 2017; Wang et al., 2017). To confirm the NMR-based identification of the BHB peak, we employed an enzymatic assay to measure BHB concentration in conditioned media from 4T07 and 168FARN (Figure 5B). The results were in perfect agreement with the NMR analysis: BHB production is significantly higher in the loser than in the winner cell clone. To test whether this metabolite was indeed responsible for the increased proliferation of 4T07, we next complemented the medium of exponentially growing 4T07 cells with purified BHB. As shown in Figure 5C, BHB increased the 4T07 proliferation rate to a level comparable to that obtained with the 168-conditioned medium. We thus conclude that loser cells increase the winner’s growth rate through the secretion of BHB.

After assessing BHB production in homotypic cell culture, we evaluated its secretion under heterotypic conditions. We grew 168FARN alone or together with 4T07 at a 1:1 ratio, maintaining the overall cell density constant. Surprisingly, despite the fact that under heterotypic conditions there are at least 50% fewer loser cells (which are the main producers of BHB, cf. Figure 5B), the overall level of secreted BHB was higher than in the homotypic culture (Figure 5D). This suggests that either the presence of 4T07 increased the production of the metabolite by 168FARN or, alternatively, that it was 4T07 that produced more metabolite when grown in the presence of 168FARN. To distinguish between these hypotheses, we cultured both lines individually for 3 days, measured BHB concentration, and then exchanged the culture medium and quantified metabolite synthesis 24 h later. The quantification of BHB produced over the last day (Day 4 BHB concentration minus Day 3 BHB concentration) shows that the 168FARN-conditioned medium had no effect on BHB secretion by 4T07 cells. In striking contrast, the production of the metabolite by 168FARN more than doubled under the influence of the 4T07-conditioned medium (Figure 5E). Thus, the winner cells stimulate the losers to produce a metabolite that boosts the former’s proliferation.

We next asked about the mode of action of BHB on the 4T07 cells. BHB can be imported by four monocarboxylate transporters of the SLC16A gene family, the expression of which varies in different cell types. We assessed the expression of each transporter by RT-QPCR and found that MCT2, MCT3, and MCT4 were barely expressed while MCT1 was highly expressed (Figure 6A) in 4T07 cells. This result suggests that MCT1 is likely responsible for the import of BHB in this cell line. Interestingly, we found that MCT1 is three times more expressed in 4T07 than in 168 cells (which, like 4T07, do not express the other MCTs – Supplementary Figure 4A), suggesting that the winner cells are more efficient at taking up this metabolite than the losers (Supplementary Figure 4B). Finally, incubation of 4T07 with BHB upregulates MCT1, consistent with a positive feedback loop that could increase the transport of this ketone body into the dominant cell line (Supplementary Figure 4C).

β-Hydroxybutyrate can be metabolized and used as a nutrient to replace glucose (Newman and Verdin, 2017). Experiments presented in Figure 2A show that fresh medium added at day 3 did not boost cell proliferation, suggesting that in this experimental setup the decrease in the carbon source is not a limiting factor for growth. It is thus unlikely that BHB is used as an energy resource to increase proliferation rate. BHB has previously been identified as an inhibitor of class I histone deacetylases (HDAC) that modulates the expression of genes involved in reactive oxygen species detoxification (Shimazu et al., 2013). Subsequently, another group found that adipocytes use BHB to modulate the expression of a subset of genes involved in the growth of breast cancer cells (Huang et al., 2017). We thus hypothesized that BHB might increase the growth rate of winners through the inhibition of HDACs, thereby modulating the expression of genes involved either in ROS detoxification or in the induction of pro-proliferative factors. In support of this idea, incubation of 4T07 cells either with 168FARN-conditioned medium or with purified BHB increased H3K9 acetylation, albeit to a lesser extent than butyrate, a bona fide HDAC inhibitor (Figure 6B).

While we could not detect in 4T07 cells any modification of expression of ROS detoxification genes reported to be regulated by BHB in other cellular models (Shimazu et al., 2013), both BHB and 168-conditioned medium led to significant transcriptional activation of interleukin-11 (IL-11) and lipocalin 2 (LCN2) (Figure 6C). Both genes have been previously described to promote cancer cell growth and to be regulated by BHB through its action on HDAC activity (Yang and Moses, 2009; Grivennikov, 2013; Huang et al., 2017). Thus, our data point to the molecular mechanisms involving direct proliferation signaling.

In addition to the positive effect of the 168FARN cells on the proliferation rate of the 4T07 clone, the data shown in Figure 2 indicate that the latter negatively influences the 168FARN growth dynamics. The NMR analysis highlighted strong lactate production (see Figure 5A). This is consistent with our observation of the media color change during culture of the two lines, indicating that the winner clone has a glycolytic type of glucose metabolism leading to a rapid medium acidification in culture. Because extracellular acidification can be detrimental for cell growth, we next asked if 168FARN were particularly sensitive to such growth conditions. We quantified medium acidification by seeding cells at different densities and measuring the extracellular pH after 3 days of culture (Figure 7A). As expected, we found that 4T07 cells acidify the medium faster and attain a lower pH during culture compared to 168FARN cells. Indeed, pH ranged from 6.94+/−0.005 (lowest density) to 6.79+/−0.003 (highest density) for the winner line and from 7.38+/−0.008 to 6.92+/−0.006 for 168FARN. To test whether 4T07-mediated extracellular acidification influenced 168FARN growth, we set up a proliferation assay for 168FARN cells grown in medium conditioned by the low and the high density grown 4T07 cells. To control for the effect of pH in the conditioned media, we included a treatment in which the medium from 4T07 was buffered at pH 7.0 by sodium bicarbonate. These experiments revealed that the medium from the low density 4T07 cells (pH 6.94) had no effect on 168FARN proliferation. In contrast, the medium from the high density 4T07 (pH 6.79) drastically decreased the 168FARN growth rate. Moreover, buffering the same medium at pH 7.0 restored the proliferative capacity of 168FARN culture (Figure 7B). We conclude that the loser clone is indeed highly sensitive to medium acidification. Taken together our data suggest that the decrease in the growth rate of 168FARN observed in heterotypic conditions is triggered by 4T07 mediated extracellular acidification.

4T07 and 168FARN were a kind gift of Robert Hipskind. All cell lines were cultured in Dulbecco’s modified Eagle medium containing 10% fetal bovine serum, 100 ng/mL streptomycin, and 100 U/mL penicillin at 37°C with 5% CO2.

For co-culture experiments a mixture of GFP-labeled and parental cells (empty-vector transduced) cells were seeded at the final density of 10⁵ cells/well in six-well plates, except where mentioned otherwise. Upon reaching confluence (3–4 days) they were harvested, diluted to the original density and replated. The remaining fraction was analyzed by flow cytometry.

Cells were lysed in boiling Laemmli buffer supplemented with protease inhibitors, then sonicated and complemented with DTT. Protein concentration was determined by BCA (Thermo Scientific) assay. Fifteen to twenty micrograms of total protein were loaded onto SDS-PAGE gels and transferred onto nitrocellulose membranes. The membrane was blocked with TBST (1× TBS with 0.1% Tween 20) + 5% milk at room temperature for 1 h and incubated with primary antibody and then with horseradish peroxidase (HRP)-coupled secondary antibody (Amersham, Piscataway, NJ, United States). Activity was visualized by electrochemiluminescence. Antibodies used in this study are anti-Histone H3 (Cell signaling Technology #9717) and anti-Acetyl-Histone H3 (Lys9) (Cell signaling Technology #9649).

Total mRNA was isolated using a RNeasy mini kit (Qiagen, Germantown, MD, United States). Reverse transcription was performed with random hexamers and M-MLV Reverse Transcriptase (Invitrogen). Real-time PCR was performed in triplicates with LC FastStart DNA Master SYBR Green I on a LightCycler rapid thermal cycler system (Roche Diagnostics, Mannheim, Germany), according to the manufacturer’s instructions. Housekeeping gene HPRT was used for normalization. Primers sequences are available upon request.

Time-lapse microscopy was performed at 37°C with 5% CO2, with images taken at 45-min intervals using an inverted Zeiss Axio-Observer microscope. The images were processed and analyzed using ImageJ software.

Cells were incubated with 10 μM EdU for 2 h, harvested and processed using the Click-iT^TM EdU Alexa Fluor^TM 647 Flow Cytometry Assay Kit (Thermo Fisher Scientific #C10424) following manufacturer instructions. Labeled cells were then analyzed on a FACSCalibur flow cytometer using CellQuest Pro software (BD Biosciences).

To determine the percentage of apoptotic cells with externalized phosphatidylserine (PS), adherent and floating cells were collected and labeled with the Annexin V-Cy3 Apoptosis Detection Kit (Abcam, Cambridge, United Kingdom, #ab14143) according to the manufacturer’s instructions. Labeled cells were then analyzed on a FACSCalibur flow cytometer using CellQuest Pro software (BD Biosciences).

β-Hydroxybutyrate concentration was measured by an enzymatic kit (Sigma-Aldrich MAK041) following the manufacturer instructions. Briefly, β-hydroxybutyrate present in the culture medium was determined by a coupled enzyme reaction, resulting in a colorimetric (450 nm) product, proportional to the β-hydroxybutyrate concentration. The absorbance was measured on a spectrophotometer.

In order to separate low molecular weight molecules from the conditioned culture medium, 5–10 ml were loaded on a Vivaspin Turbo 15 PES, 3,000 MWCO column (Sartorius VS15T91) and centrifuged at 4000 × g for 30 min following the manufacturer instructions. Both fractions were then used for subsequent experiments and RMN analysis.

NMR experiments were recorded at 293 K and pH 7 on an AVANCE III BRUKER spectrometer operating at 700 MHz (proton frequency), using a Z-gradient shielded TCI 1H-13C-15N cryoprobe. Fully relaxed 1D 1H spectra were aquired with the regular 1D NOESY, using 5 s as relaxation delay. The samples consisted on 1.5 mL of cell media (fresh or conditioned by cell culture), lyophilized and dissolved in 500 μL of deuterated phosphate buffer (50 mM, pH 7). DSS (EURISOTOP^©, final concentration: 0.5 mM) was added as internal reference for chemical shift referencing and as a concentration standard for spectra normalization. The assignment of the 1H resonances of the compound of interest in this study (lactate, β-hydroxybutyrate) was based on chemical shifts reported on the literature (1) and further confirmed using 2D [1H, 1H] (TOCSY) and [1H-13C] (HSQC, HSQC-TOCSY) NMR spectroscopy.

Experiments were repeated at least three times. Data are presented as mean ± SEM. An Independent Student’s t-test was performed to analyze the assay results; a two-tailed Student’s t-test was used to compare the intergroup differences. Significance was accepted for values where P ≤ 0.05 (^∗), P ≤ 0.01 (^∗∗), P ≤ 0.001 (^∗∗∗).

Our aim is to determine the general nature of the evolutionary dynamics in a form that can be readily compared to other systems (as opposed to generating quantitative predictions for our particular system). Accordingly, we chose to fit a simple, standard model to each distinct phase of the dynamics, such that the inferred parameter values have straightforward ecological interpretations. A key advantage of our method is that it is generic; in principle, the same method can be applied to any experimental evolution set-up with two competing populations of cancer cells, bacteria, or other entities.

This mathematical approach is in the same vein as that of Kaznatcheev (2017) and Kaznatcheev et al. (2019) but with three important differences. First, our method can accommodate a smaller data set and is thus more economical because we mostly rely on measurements of initial and final proportions in competition assays, such as can be determined via flow cytometry, rather than extensive time-lapse image analysis. Second, whereas Kaznatcheev (2017) and Kaznatcheev et al. (2019) confine their analysis to exponential or logistic growth phases, we also examine phases in which cell populations exhibit non-logistic dynamics. Third, because we consider non-logistic growth phases, we use a density-dependent rather than a frequency-dependent model.

We note that to make quantitative predictions of outcomes in different scenarios, we would require a different type of model with equations describing the dynamics of paracrine factors mediating clonal interactions. This more complicated model would include several more parameters and design choices (for example, how each paracrine factor’s production rate and its effects vary with its concentration) and would thus be non-identifiable in the absence of detailed paracrine concentration measurements. Obtaining such measurements remains as a challenge for future studies.

We define the intrinsic growth rate as the exponential growth rate in the absence of interactions. In the Lotka–Volterra differential equations, this parameter is multiplied by the population size of the respective type. The intrinsic growth rate is the limit of the net growth rate as the population sizes approach zero (when interaction terms are negligible).

We define the net growth rate as the actual rate of change of the population size (i.e., the time derivative), which is the sum of the basic growth rate and interaction terms.

Supplementary Figures 6, 7 illustrate some of the mathematical relationships relevant to our methods.

We describe the exponential phase 1 dynamics as

where L (loser) and W (winner) are the population sizes of 168 and 4T07, respectively, and r_L,1 and r_W,1 are the respective growth rates.

In phase 2, we assume a density-dependent competitive Lotka–Volterra model, parameterized in terms of intrinsic growth rates r_L,2 and r_W,2 and interaction terms a, b, c and d:

In the homotypic case, terms bW and cL vanish and the phase 2 model is equivalent to logistic growth. We combine the two models and fit to the normalized time-lapse data for the homotypic growth curves using least-squares with R package deSolve (Soetaert et al., 2010) to infer the values of r_L,1, r_W,1, r_L,2, r_W,2, a, and d.

In phase 3, we assume the same model as in phase 2 except we replace r_W,2 by r_W,3 to account for the change in the 4T07 net growth rate (equivalent to adding a density-dependent death rate).

To infer the interaction parameters b and c we need data that covers a wide range of proportions of the two cell types. Since our time-lapse data is limited to only a few initial conditions, we fit the model to the outcomes of serial competition assays, and we employ the heterotypic time-lapse data for validation only. First we define

The time derivative of the s is then equal to the net growth rate difference, which in phase 2 is

In the limit w→1, the final term (c−a)L is negligible and we can obtain b in terms of dsdt, W, and parameters whose values we have already inferred, as follows:

To obtain W, we note that in the limit w→1,

which is the logistic differential equation with solution

where r = r_W,2 and t₁ is the time at which phase 2 begins. We can thus use our previously inferred parameter values to obtain W(t) at every time t in phase 2 (note that if there were not an analytical solution then we could have solved the equation numerically).

Since W and dsdt are linearly related, we can replace them by their mean values:

Using the mean values to calculate b is convenient as our competition assays reveal only the initial and final values of s. Specifically, we take the means in the interval [t₁,t₂], where t₂ is the time at which phase 2 ends and

It remains only to obtain the value of the above expression – known as the gain function – in the limit w(t₁)→1. From competition assay data, we can immediately obtains(t2)=log⁡w(t2)1-w(t2) for each value of s(0)=log⁡w(0)1-w(0). To infer w(t₁) and s(t₁), we need to adjust for the exponential growth of both cell types during phase 1:

We thus obtain the values of s(t₁) and w(t₁) in each competition assay. Finally, we determine by linear regression the relationship between △s/△t and w(t₁) (Supplementary figure 5B) and, from the equation of the regression line, infer the value of △s/△t in the limit w(t₁)→1. We then have everything required to infer the value of b. By an analogous method (switching L and W, b and c, and a and d) we also infer the value of c.

In our regression to determine the relationship between △s/△t and w(t₁), we excluded data from the first round of competition assays (days 0–3 in Figures 2B,C) because these measurements were unusually variable, and this variance was most likely an experimental artifact. Specifically, setting up the initial experiment took substantially longer than carrying out subsequent replatings as additional steps were required before seeding the cells. Since cells were kept for longer in suspension before the first round, they will have experienced more stress and potentially mortality. This means that results of the first round of competition assays are likely to be less reliable than results of subsequent rounds. For completeness, Supplementary Figures 5C,D show linear regression applied to the entire data set, including the first round.

To find carrying capacities, we note that the phase 2 model can alternatively be parameterized as

where the parameters are calculated as in Table 1. The carrying capacities K_W and K_L are the upper limits approached by the population sizes of W and L, respectively, during phase 2.

In a growing tumor, we expect cell–cell competition to be less than in our in vitro experiments, because, in the former, resources are continually replenished and waste materials removed by the host circulatory system. The evolutionary dynamics will then mostly depend on the difference in intrinsic growth rates and interactions mediated by diffusible factors. Furthermore, during tumor growth, the dynamics may be better described by a frequency- rather than a density-dependent model. We can then describe the evolutionary dynamics within the framework of evolutionary game theory using the payoff matrix

where α_L,α_W < 0 denote the harm inflicted by W on L and W, respectively; β_L,β_W > 0 are the benefits bestowed by L to L and W, respectively; and γ > 0 is the difference between the intrinsic exponential growth rates. The relative values of the entries in the payoff matrix determine which game (for example, prisoner’s dilemma or hawk–dove) is equivalent to the evolutionary dynamics.

The parameter values inferred for phase 2 of the competition assays imply

in which case the evolutionary dynamics are equivalent to a prisoner’s dilemma game for which W is the only evolutionarily stable strategy (ESS). This means that W (4T07) can invade and stably replace a population of L (168).

If instead α_L−γ > α_W then the payoff matrix defines a hawk–dove game that permits coexistence. In this scenario, W harms itself more than it harms L, and this difference outweighs W′s higher intrinsic growth rate. This could happen, for example, if harmful factors produced by W imperfectly diffuse, so that W cells experience a higher concentration than L cells. At the mixed ESS, the W proportion is

However, if additionally β_L−β_W > γ (so that L benefits itself more than it benefits W, and this difference outweighs W′s higher intrinsic growth rate) then coexistence again becomes impossible as the game again becomes a prisoner’s dilemma but with L as the ESS.

In a resource-poor environment, we might describe the evolutionary dynamics using the payoff matrix

where δ is the reduction in W′s intrinsic growth rate due to the degraded environment (as inferred for phase 3 of our 96-h competition assays). This scenario favors L and suggests that L may be the ESS in a resource-poor environment, such as hypoxic regions within a tumor.