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Biological experiments
Cell culture
We obtained unlabelled parental (sensitive) and radiation-resistant populations from two prostate cell lines, PC3 and DU145, from the Liu laboratory (University of Toronto, Canada). Radioresistant cell populations comprised pooled cells from the parental line that survived a clinically-relevant course of radiotherapy23,24. To produce stable, fluorescent cell lines, we transduced cells using lentiviral particles containing the vectors, pCDH1-CMV-GFP-EF1-Hygro or pCHD1-CMV-DsRed-EF1-Hygro (Systems Biosciences), collected the top 30% of brightest cells by flow cytometry, and used hygromycin B (50 mg/mL, Gibco) for selection (200 µg/mL for PC3 and 250 µg/mL for DU145). Cell lines were cultured in DMEM medium (low glucose, pyruvate, GlutaMAX, Gibco) supplemented with 25 mM HEPES (Gibco), 10% foetal bovine serum (Sigma or Pan-Biotech), and 1% penicillin/streptomycin. Authentication was performed using STR profiling (Promega PowerPlex 21 PCR kit, Eurofins), and mycoplasma checks were performed routinely using MycoAlert Mycoplasma Detection Kit (Lonza). All cells were maintained in an incubator (37 °C, 5% CO2). Unlabelled cells were cultured for up to 10 passages (~6 weeks) for transduction; labelled cells were cultured for up to 10 passages (~6 weeks). No additional courses of radiation were used to maintain resistance. We measured clonogenic survival at 2 Gy (SF2) to verify that the labelled cells maintained the resistance phenotype for up to 12 passages (~8 weeks) in culture.
Monolayer growth experiments
Single cells (1 × 103 cells/well, 200 μL medium) were seeded in triplicate in flat-bottom 96-well plates, allowed to attach overnight, irradiated (0, 2, or 6 Gy), and imaged daily using brightfield (Incucyte Live Cell Imaging System, Sartorius). The medium was changed every 2 days. Cell confluence was determined using Incucyte Base Software (Sartorius).
Clonogenic assays
Survival of cell lines after radiation was measured using a clonogenic assay36. Briefly, cells were seeded in triplicate in six-well plates and irradiated using a Cs-137 (dose rate of 0.89 Gy/min) or an X-ray irradiator (195 kV, 10 mA). Surviving colonies were stained after 10 days with crystal violet and counted. The surviving fraction was calculated as (number of colonies/number of seeded cells) × plating efficiency.
Response to cisplatin
To check whether the RR cells had altered DNA damage response, we measured the cell viability of each population in response to a range of concentrations of cisplatin using a modified cytotoxicity assay37. Briefly, single cells (2 × 103 cells in 100 μL/well) were seeded as monolayers in triplicate in a 96-well plate and allowed to attach for 36 h before treatment. Increasing concentrations of cisplatin (made in 100 μL/well) were added to each well resulting in a final volume of 200 μL/well; cisplatin (Sigma) was prepared fresh for each treatment by dissolving the powder in 0.9% sterile-filtered saline to a stock concentration of 3.3 mM. Treated cells were then cultured for 72 h in cisplatin before they were fixed with 10% formalin. Cell confluence was determined using Incucyte Base Software (Sartorius) and reformatted to a concentration–response curve by normalising cell confluence values to the untreated well.
Spheroid generation and culture
Homogeneous and mixed spheroids were generated in 96-well, ultra-low attachment plates (7007, Corning) by seeding different ratios of parental and radioresistant cell populations (2 × 103 total cells/well) using Matrigel (5% v/v, Corning) to promote spheroid formation38. For all spheroid experiments, after a formation phase of 3 days, spheroids were fed every 2 days by replacing 50% of the medium in each well with fresh medium (200 μL total/well). Culture medium and incubation conditions were as described under ‘Cell culture’. Spheroid volumes were calculated using SpheroidSizer39.
Unirradiated spheroid growth experiments
Spheroids were generated as described in ‘Spheroid generation and culture’ and monitored for growth by brightfield imaging (Leica DM IRBE, Hamamatsu).
Flow cytometry
The proportions, survival, and cell cycle of each population from spheroids were measured by flow cytometry. Mixed unirradiated or irradiated spheroids (seeded 1:1 parental:RR, 6–8 pooled/group) were incubated with EdU (10 µM final concentration) 12 h prior to dissociation, dissociated (100 μL Accumax, Millipore) for 20 min at 37 °C, washed with phosphate-buffered saline (PBS), centrifuged (300 × g, 5 min), and incubated with efluor-780 (1 μL/mL PBS; ThermoFisher Scientific) for 30 min on ice in the dark to distinguish live/dead cells. After washing in PBS, samples were fixed for 10 min in IC Fixation Buffer (ThermoFisher Scientific), and permeabilized and stained with Click-iT Plus EdU Alexa Fluor 647 (ThermoFisher Scientific) according to manufacturer’s instructions. Following a wash in 1× saponin, cells were incubated 30 min with FxCycle Violet Stain (1:1000, 300 µL of 1× saponin; ThermoFisher Scientific) before being run on the BD LSR Fortessa X-20 Cytometer or the Attune NxT Flow Cytometer using the 405, 488, 561, and 633 lasers. Data were analysed using FlowJo (Treestar, Inc.) as described in Supplementary Figs. 2 and 4.
Spheroid growth experiments after radiation
To determine bulk radiation response of spheroids, PC3 cells were seeded as spheroids (n = 15 per dose per group) with 4 groups as described in ‘Spheroid generation and culture’: parental, 9:1 parental:RR, 1:1 parental:RR, and RR. After formation, spheroids were irradiated (0, 2.5, 5, 7.5, 10, 15, and 20 Gy) on day 4 and imaged for up to 48 days to monitor regrowth using brightfield (Celigo Imaging Cytometer, Nexelcom). After log-transforming the volume data, we calculated the radiation-induced growth delay (days) relative to untreated spheroids as the time for each irradiated spheroid to reach a volume endpoint (2.5 times the starting volume right before irradiation); we selected the lowest endpoint that was still within the exponential growth phase of all spheroids in the experiment. The average time for untreated spheroids to reach endpoint was estimated in R by local regression using the loess function with “direct” surface estimation to allow extrapolation for the parental spheroids (R project, v. 3.6.2).
Regrowth experiments were repeated by irradiating day 3 spheroids from three groups (parental, mixed and RR) of PC3 cells (6 Gy, n = 17–18/ group) and of DU145 cells (6 Gy, n = 12/group; 10 Gy, n = 20/group). Spheroids were imaged using brightfield (Leica DM IRBE, Hamamatsu) for up to 27 days (PC3) and up to 23 days (DU145). The radiation-induced growth delay (delays) was calculated as above, but with different endpoints (3.5 times starting volume for PC3 and 4 times starting volume for DU145) to ensure the endpoint was within the exponential growth phase. Data from Fig. 2 were used to estimate average time of untreated spheroids; the ‘span’ parameter of the loess function was reduced from the default of 0.75–0.5 for the unirradiated DU145 spheroids to better estimate the average time of reaching the endpoint.
To measure changes in the radiation response of PC3 cell populations within spheroids, untreated homogeneous and mixed spheroids were grown until day 5 or 11, dissociated using Accumax, seeded as single cells for clonogenic experiments, and allowed to attach for 6 h prior to radiation. Fluorescent colonies were counted using the Celigo Cytometer.
Immunofluorescence
For immunofluorescence and H&E experiments, spheroids were treated and fixed prior to staining40. To investigate the spatial distribution of fluorescent populations, sections were hydrated in PBS, stained for 10 min with Hoechst (1 μg/mL in PBS, Sigma) to visualise nuclei, and mounted using ProLong Diamond Antifade Mountant (ThermoFisher). For hypoxia, spheroids were pre-treated with 300 μM of the hypoxia drug EF5 (gift from Dr. Cameron Koch, University of Pennsylvania) prior to fixation. They were then permeabilized (PBS containing 0.3% Tween-20, 10 min), blocked (5% goat serum in PBS containing 0.1% Tween-20, 30 min), stained using anti-EF5 antibody (75 μg/mL; from Dr. Cameron Koch) overnight at 4 °C, washed (ice-cold PBS containing 0.3% Tween-20, 2 × 45 min)40, stained for nuclei as above, and mounted. For Ki67, spheroid sections were permeabilized (PBS containing 0.3% Tween-20, 10 min), blocked (5% goat serum in PBS containing 0.1% Tween-20, 30 min), and incubated overnight at 4 °C with primary antibody (clone SP6, 1:100, Vector Laboratories). After washing in PBS, sections were incubated for 1 h with goat anti-rabbit Alexa Fluor 647 (4 μg/mL, ThermoFisher), washed, and stained with Hoechst 33342 (5 μg/mL, Sigma) for 10 min. Slides were mounted and imaged using epifluorescence microscopy (20× objective; 0.30 NA; 0.64 μm resolution; excitation lasers: 395, 470, 555, and 640; Nikon Ti-E). Sections were stained using H&E and imaged using a Bright Field Slide Scanner (Aperio CS2, Leica) to visualise necrosis. To quantify the ratio of parental to radioresistant populations in spheroid cross-sections (n = 16 spheroids from 4 batches), we measured the number of pixels from each population (i.e., signal) by applying a threshold value 5 times higher than the median value of the background (i.e., noise) (Octave 4.4.1).
Oxygen consumption measurements
OCR was measured from each population using the Seahorse assay. Cells (1.2 × 104/well) were seeded in triplicate using the normal culture medium in a Seahorse XF 96-well microplate (Agilent) and allowed to attach overnight. Prior to the assay, cells were washed with and incubated in assay medium (DMEM basal medium containing 5 mM glucose, 4 mM glutamine, 5 mM pyruvate, pH 7.4; 200 μL/well) for 2 h at 37 °C without CO2 to degas the medium. Calibrant buffer (200 μL/well) was added to wells of the probe plate and also left at 37 °C without CO2 to degas. After OCR was measured on the Seahorse XF Analyser (Agilent Biosciences), cells were fixed using 4% paraformaldehyde, stained using Hoechst 33342, and counted (Celigo Cytometer, Nexelcom).
Transwell experiments
Co-culture experiments were performed to measure whether transferred factors between cell populations enhanced survival under hypoxia. Cells were seeded in triplicate (3.0 × 104/bottom well and 1.0 × 104/insert) in 12-well plates and in Transwell inserts, and allowed to attach overnight. Once the medium was changed, the plates were placed into normoxia or hypoxia (0.1% O2) for 24 and 120 h. Cells were fixed using 4% paraformaldehyde, stained with Hoechst (5 μg/mL), and counted (Celigo Cytometer, Nexelcom).
Statistics and reproducibility
Data were evaluated for equal variance using homoscedasticity plots (absolute value of residual vs predicted value) and for normality using Q–Q plots (Prism 8.0, GraphPad). Unless otherwise indicated, statistical significance was evaluated using one-way ANOVA, two-way ANOVA, or a mixed-effects model followed by multiple testing correction (α = 0.05). For clonogenic assays, the radiation protection factor was calculated as the area under the dose–response curve (AUC) for the RR cell populations divided by that of the parentals; AUC values were analysed for significance using a Student’s t test (unpaired, one-tailed, α = 0.05). For cisplatin cytotoxicity assays, IC50 values were calculated using a normalised response, variable slope, dose–response model (Prism, 8.0, GraphPad) and evaluated for statistical significance using extra sum-of-squares F-test. For post-radiation growth experiments, survival curves were analysed using the Mantel–Cox (log-rank) test and adjusted for multiple testing using Holm’s correction; spheroids that did not reach the endpoint during the timeframe of the experiment were marked as ‘censored’ on the final day of the experiment (please see Supplementary Methods section 3 for further details). Due to heteroscedasticity, cell counts from flow cytometry experiments involving cell cycle and death, and from co-culture Transwell assays were analysed using overdispersed Poisson or binomial regression models (please see Supplementary Methods section 3 for further details). For quantification of population proportions in microscopy images, pixel numbers were analysed using a two-tailed, Wilcoxon matched-pairs signed-rank test. Adjusted P values (Padj) are reported in the main text for experiments where multiple comparisons were performed.
Data points represent biological replicates; experiments were performed using at least two separate batches of cells. We note the following data exclusions: missing data from some time points due to technical failures in imaging (Fig. 1), one excluded mixed DU145 spheroid because its growth did not resemble that of the other 35 spheroids (Fig. 2a), and one excluded plate of PC3 spheroids from survival analysis (Fig. 4) because of irregular growth that did not match the other nine plates. Sample sizes were approximated using effect sizes from pilot studies to ensure power (approximate β = 0.8); randomisation and blinding were not possible.
Mathematical experiments
Non-spatial mathematical models
We used the logistic growth model to describe the growth of homogeneous tumour spheroids26. Thus, the rate of change of spheroid volume V at time t is given by

$$frac{{dV}}{{dt}} = rV left(1 – frac{V}{K}right),$$
(1)

where r  > 0 represents the growth rate, K  > 0 is the carrying capacity (the limiting volume of the spheroid) and V(t = 0) = V0 denotes the spheroid volume at t = 0. The analytical solution to the logistic model is given by

$$Vleft( t right) = frac{{V_0Ke^{rt}}}{{K + V_0(e^{rt} – 1)}}.$$
(2)

The Lotka–Volterra model was used to describe the growth of mixtures of parental and RR cell populations

$$left. {begin{array}{*{20}{c}} {frac{{dV_P}}{{dt}} = r_PK_Pleft( {1 – frac{{V_P}}{{K_P}} – lambda _{RR}frac{{V_{RR}}}{{K_P}}} right)} \ {frac{{dV_{RR}}}{{dt}} = r_{RR}K_{RR}left( {1 – frac{{V_{RR}}}{{K_{RR}}} – lambda _Pfrac{{V_P}}{{K_{RR}}}} right)} end{array}} right},$$
(3)

with VP(t = 0) = VP0 and VRR(t = 0) = VRR0. In these equations, VP and VRR represent respectively the volumes of parental and RR populations, rP and rRR their initial growth rates, KP and KRR their carrying capacities, and VP0 and VRR0 their initial volumes. The parameters λP and λRR describe the effect that parental cells have on RR cells, and vice versa. These type of interactions, found in ecology12, may be competitive (λP  > 0 and λRR  > 0), mutualistic (λP  More

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