Figure 1.
Outline of the experimental/computational method.
(A) Individual cancer cells respond to the challenge of a treatment activating molecular pathways that cause cell cycle arrest, damage repair or cell death. The response has complex time-dependence and the effects are still detectable in the descendants of the cells exposed to the drugs. Cell outcomes are not homogeneous, and the overall antiproliferative response at the cell population level is the sum of the different stories of all cells. (B) Different experimental techniques can be applied to interrogate the biological system and retrieve information about cell proliferation during or after treatment. Flow cytometry gives percentages of cells in the various cycle phases, while time-lapse live cell microscopy indicates the propagation of the effects through subsequent cell generations. A proper experimental plan, including time course measures with both techniques on samples treated with different doses, can potentially give a complete scenario of the effects in play, but is not easy to interpret. (C) A computer model renders in silico the dynamics of cell proliferation, based on parameters associated with unperturbed growth and the activity of cell cycle checkpoints, producing outputs that mimic experimental data obtained with both FC and TL platforms. A best fit rendering fully consistent with all experimental data discloses the details of the proliferation and of the underlying checkpoint activities, with their dose-dependence.
Figure 2.
Experimental plan and first-level data analysis.
For FC analysis, cells from replicated samples were collected, fixed and stained at 6, 24, 48 and 72(A) Representative FC data: i) monoparametric cell cycle analysis, from which we calculated %G1, %S, %G2M, ii) biparametric DNA-BrdU analysis after BrdU pulse labeling, from which we calculated the percentages of BrdU-positive cells (%BrdU+), i.e. cells which were in S phase at the time of labelling, and undivided BrdU+ cells (%Und+). (B) Representative TL data: movies were collected and each cell was tracked with its descendants. At least 300 lineages were analysed in each treatment group. Three representative lineages are shown, where coloured squares indicate the outcome event of each cell. Analysis of lineage data gave population statistics like frequency distribution of intermitotic times and number of cells in each generation (bottom).
Figure 3.
Structure of the modelling framework, with detail of cell progression and perturbation modules.
(A) The model reproduces the flow of cell cohorts through the cell cycle phases and subsequent generations, each phase including specific quiescence (Q) and perturbation modules. A cohort of cells entering a phase is first processed by the quiescence module, committing a fraction of them to the G1 quiescence compartment. Then untreated cells progress through the subsequent age compartments of each phase, exiting at different ages as shown in Figure S4 and detailed in supplementary Text S3. In treated samples this process is altered by perturbation modules, which can be applied to any phase, providing a flexible framework to build proliferation models with the desired complexity. (B) Cycling cell death module, exemplified for G1 phase. It applies first-order death kinetics to all cycling cells in phase G1 with a rate DRG1. (C) The mitosis/polyploid module acts on dividing cells, doubling the number of cells exiting G2M and assuming that a fraction (pPol) of newborn siblings re-fuse, collecting re-fused (polyploid) cells in a separate compartment, from which they die with a rate DRPol. (D) Checkpoint modules, exemplified for G1 phase. A specific type of checkpoint module can be selected, as described in Computational Methods, with parameters DelG1 (producing a transit delay), pBLG1 (block probability), DRBLG1 (death rate of blocked cells), RecG1 (recycling rate of blocked cells).
Figure 4.
Data and fit with the final model.
Time courses of measurable quantities obtained from the final model compared with experimental data (symbols), for each radiation dose. The good quality of the fit indicates that the model successfully predicts FC and TL data of all doses at the same time. a) Time course of %G1, %S, %G2M; b) percentage of residual undivided BrdU+ cells (supplementary Text S1); c) number of cells in gen0 (g0), gen1 (g1), gen2 (g2), gen3 and higher generations (g3+) and polyploid (pol), normalized assuming N(0) = 1000; d) percentage of cells which died in the 0–72 h observation time among cells entered in each generation. The symbols and error bars represent the mean and standard deviation of experimental data of at least three independent experiments (FC) or five replicate culture wells (TL).
Figure 5.
Dynamic rendering t of proliferation after X-ray exposure.
Cell distributions in the cell cycle and over generations are shown at representative times (24 h and 72 h) and doses (ctrl, 0.5 and 5 Gy). The whole time courses for all doses (the final best fit model) are reported as Supplementary Videos. The plots shows the density of cycling cells in G1 (red dots in the inner sector), S (green, intermediate sector) and G2M (blue, external sector) phases according to their age (progressing clockwise from the starting point of each phase). The whole cycle lasts 24 h. The S and G2M starting points are placed in the figures at points corresponding to the mean duration of the previous phase, and are preceded by a compartment collecting quiescent cells. G1 blocked cells are presented as red dots in the intermediate sector, before the starting point of S phase, G2M-blocked cells are blue dots at the end of the cycle, external to the G2M sector. Polyploid cells are presented as black dots in a sector placed to the upper right of the cycle.
Figure 6.
Checkpoint activities in the best-fit final model, as a function of the dose, in subsequent generations.
(A), G1- and G2M-block probability (i.e. the fraction of cell intercepted and blocked at G1 and G2M checkpoints among cycling cells entering these phases) in irradiated cells (gen0) and their descendants (block probabilities in gen1 and gen2 were not distinguishable). (B) Outcome of G1-blocked cells, showing the percentages of cells that re-enter the cycle, die or remain blocked at 72 h in the indicated generations. (C) Outcome of G2M-blocked cells, symbols as in panels B. (D) Dose-dependence of the delay in phase S (fractional reduction of DNA synthesis rate) in gen0 and gen1. Data fit required a distinction between the delay of cells irradiated in S phase (BrdU+) and in G1/G2M (BrdU−) in gen0. No delay was found in gen2. (E) Polyploidization rate, as the percentage of cells that re-fused in gen1 and gen2. Error bars indicate 95% confidence intervals for parameter and derived quantities (e.g. fraction of blocked cells), calculated by fitting 1000 synthetic datasets generated by a Monte Carlo procedure (see Uncertainty Analysis in supplementary Text S4).
Figure 7.
In silico experiments, exploring the consequences of default of specific checkpoints.
(A) Simulated growth curve in the absence of perturbations in descendants (red line) compared to the best fit final model (black line) and proliferation in untreated cells (green line). (B) Growth curve in the final model (black line), in the presence of an additional agent (drug A) increasing the probability of G1 block (p = 0.8) (red line) and with an additional agent (drug B) preventing exit from G2M block (green line).