Mixture models to investigate the presence of multiple cellular populations before cisplatin, with different cycling kinetics

While full lineages were tracked in the HCT116 dataset [18], only one cell per sister-pair was tracked in the U2OS dataset [37]. Hence for the latter, we resorted to using a mixture model to check whether multiple subpopulations with distinct proliferation rates could be identified from the pre-cisplatin cells. Since previous work has identified the Exponentially Modified Gaussian (EMG) as a good model of the IMT distribution [18], we used a mixture of EMGs for this purpose. The EMG is a convolution of an exponential and Gaussian distributions, the former parameterized by λ and the latter by μ and σ. We used a mixture of two EMGs to fit the IMT distribution data and compared with the fit to a single EMG using the Akaike Information Criterion (AIC). Details of the likelihood function and inferred parameters can be found in Sect 3 of S1 Text.

Predicting population dynamics from single-cell measurements

Age structured population growth models link the net proliferation rate of the population to single-cell division and death time distributions [38–41]. They relate the exponential proliferation rate of the population to the age dependent division and death rates (hazards) via the Euler-Lotka equation:

(1)

where , is the net hazard, and b(y) and are the age dependent birth and death hazards respectively. γ is the unique root to Eq 1 that gives the population growth rate. The hazard function is a time-dependent generalization of the rate parameter in exponential distributions, given by the ratio of the probability density and survival function (see Sect 8 in S1 Text for details). Before drug administration, the distribution of intermitotic times (IMT) of the cells was fit to an Exponentially Modified Gaussian (EMG) using Maximum Likelihood Estimation (details in Sect 8 in S1 Text). This provided the IMT probability density function, which in turn allows calculation of the survival function and hazard at any given instant in time. The same analysis was performed post drug on IMT and AT (apoptosis time) distributions to get the hazards for division and death respectively. The total hazard (for any event) post drug was then simply calculated as the sum of division and death hazards. The hazards were then used to solve the integral Eq 1 to obtain the population growth/decay rate γ. Detailed descriptions of parameter inference and calculation of the hazard and proliferation rates are provided in Sect 8 in S1 Text.

Stochastic simulations of age-structured population dynamics – Model M0

Model M0 corresponds to the conventional age-structured model of population dynamics which generalizes constant division and death rates to age-dependent rates, giving rise to exponential growth at long times. Stochastic simulations of this model were performed for both pre-drug (only division) and post-drug (division and death) scenarios (details can be found in [18,39]. In brief, single-cell distributions of division and death times were taken from the experimental datasets and converted to age-dependent birth and death rates or hazards. For pre-drug simulations, each new cell in a lineage was assigned an age 0 at birth and for each following time-step, the probability of dividing was computed based on the age-dependent division hazard at that time. For post-drug cells with the additional fate of cell death, the total age-dependent hazard at any time was given by the sum of division and death hazards. These hazards were used at each time-step to compute the probabilities of dividing or dying for each extant cell independently. To compute the probabilities, the hazards were used as follows: the probability of either event (division or death) happening in a time step was calculated using the net survival probability , where , denote the post-cisplatin division and death hazards of a cell respectively, and , denote the parameter vectors parameterizing the post-drug IMT and AT distributions respectively. Once an event was destined to occur, the choice between division or death for that cell was generated based on the ratio . For cellular events before addition of the drug, only the hazard function was used, where denotes the parameter vector of the pre-drug IMT distribution. The time step was chosen such that it was much smaller than the average times of the IMT and time to death distributions. At each division event, two new cells were stored as nodes in the extant cells list and the graph that was uniquely defined for each lineage tree. Attributes like ‘birth time’, ‘state’, ‘fate’, etc. were stored in these newly created nodes for future reference. Further details can be found in Sects 8 and 9 in S1 Text.

Competing risks analysis of exponentially distributed division and death time distributions

Since cells exposed to cisplatin can undergo either division or death (apoptosis), these events are mutually exclusive and hence give rise to ‘competing risks’ for each cell. A well known problem in statistics [42], the competing risks effect leads to significant biases in the experimentally measured times to division and death after cisplatin addition [18]. Since only the event with a shorter time will ultimately be observed for any single-cell, the measured division and apoptosis times will be biased towards lower values, thereby hiding the actual underlying distributions of division and apoptosis times. These underlying distributions therefore are not directly measurable, but need to be inferred from the measured (but biased) division and apoptosis times. A simple analytical calculation using exponential waiting times for the birth and apoptosis events illustrates the competing risks problem, and demonstrates how inference of the underlying distributions can be carried out (see Sect 7 in S1 Text for details). If the underlying independent probability densities of times to mitosis and apoptosis are given by and respectively, then the corresponding measured distributions will have faster decaying exponentials, exhibiting the bias towards shorter times (Sect 7 in S1 Text). From a single cell microscopy experiment measuring these biased waiting times t_i and the number of mitosis and death events (N_m and N_a respectively), it can be shown that accurate estimators of and – parameters of the unbiased distributions, are given by: and (see Sect 7 in S1 Text for details).

A Markov Chain Monte Carlo approach to competing risks analysis for HCT116 and U2OS cells.

While exponential distributions allow for an analytical solution of the competing risks inference problem, non-exponential distributions along with the pre and post-drug scenarios in the HCT116/U2OS experiments complicate the inference problem and preclude analytic solutions. For the post-drug scenario we needed to find the underlying distributions of IMT and AT as mentioned in the competing risks section, which are modeled as EMGs having three parameters each (μ, σ and λ). We performed the inference of these distributions from the biased IMT and AT values obtained from the experiment using a Metropolis-Hastings Markov Chain Monte Carlo approach. In this method we sampled a total of 8 parameters from their posteriors. Six of these parameters describe the two distributions, IMT and AT (μ, σ and λ for each distribution). The remaining 2 parameters describe the probability of the cell to enter a quiescent stage and the delay time for the drug to act. To find the posteriors of these parameters, we formulated likelihoods for all the division and death events that are possible post-drug and assumed a uniform prior to obtain the posterior using Metropolis Hastings MCMC. Details of the inference method are provided in Sect 8 in S1 Text.

Beyond age-structured populations: Stochastic simulations of Models M1, M2 and M3

Model M1: In Model M1, we defined the cells to exist in one of two states before drug administration, one being susceptible to die (sensitive, S’) while the other persists (pre-DTP, P’) in the face of drug treatment. The cells were allowed to interconvert between these states in drug naive conditions. S’ cells at time of cisplatin addition were assigned the death fate with 100% probability. Cells in the P’ state at time of cisplatin were assigned fates to divide or survive (without dividing), with some probability for each fate. The times taken for the division or death events were sampled from the respective distributions measured in the experiments. This model could recapitulate the experimentally observed lineage correlations and biphasic exponential decay, but not the drug-concentration dependence of the decay rates.

Model M2: In this model, the cells did not possess any ‘states’ before cisplatin administration. As a result, at time of drug addition, every cell was statistically identical, and received randomly assigned fates (die, divide or survive without dividing) with probabilities that depended on the drug concentration. Higher probability of death was assigned for higher drug concentrations. While this model could explain the population decay rates observed in the experiment as well as biphasic decay, it failed in explaining the correlations observed in the final post-cisplatin fates of lineage related cells.

Model M3: This model is essentially a combination of Models M1 and M2, where we simulated cells in two states S’ (sensitive) and P’ (pre-DTP) prior to drug addition with inter-state transitions. After the drug, the cells were assigned fates from 3 possibilities – death, survival or division, with probabilities conditioned on their state at the time of drug treatment. The state-dependent fate probabilities were chosen from a fate-matrix F. If the fate was division or death, the waiting time to the fate was generated by using either the experimentally measured post cisplatin division or death hazard respectively, using the probabilistic framework described above for simulating Model M0. Note that unlike in M0, here only one appropriate hazard is used post-cisplatin because of the fate assignment using F, and no competition between events occur. If a cell was assigned division as the fate, the two resultant daughters maintained the original state of the mother cell at time of drug addition, and the fate assignment process was repeated using the same matrix F. The entries of F that gave the closest match of simulations with experiments were: for state S’ – , and while for state P’ – , and . This model could quantitatively recapitulate both the lineage correlations as well as the drug concentration dependent decay rates measured in the experiments. Further details are provided in Sects 9 and 10 in S1 Text.

Calculation of Barcode abundance and diversity from Model M3

Model M3 was used in the barcoding simulations with parameters given above. To mimic barcodes, we simply provide lineage numbers to the initial ancestor cells that the simulations of Model M3 are started with. These lineage numbers are then maintained across cell divisions, mimicking the inheritance of barcodes. To calculate the barcode abundance, the frequencies of each of the barcodes were extracted just before cisplatin addition as well as 72 hours post cisplatin exposure.

Let n_i be the number of times the i^th barcode is observed. Then the probability p_i of observing the barcode is given as

(2)

The Shannon Diversity Index (SDI), which gives a quantitative measure of barcode diversity, was then calculated as

(3)

The difference in the SDI before and after drug then provides the change in diversity of barcodes. This procedure was repeated 100 times to generate error bars in the change in SDI. Further details are provided in Sect 11 in S1 Text.

Luria-Delbrück analysis of HCT116 dataset and Model M3

For both the HCT116 dataset and simulated lineages from Model M3, extant cells at time of drug administration were identified. These cells were then followed to determine their end fate at the last time frame of the experiment (survival versus death). If a cell divided post drug, then randomly one daughter was chosen. As a result, the number of cells whose fates were identified equaled the number of cells extant at the time of drug addition. From these cells, lineages were identified and number of persisters per lineage was determined to compute the index of dispersion (variance by mean). This procedure was repeated on the same dataset (originating either from the HCT116 experiment of Model M3) 100 times to generate error bars on the index of dispersion.

Subpopulations with distinct proliferation rates cannot be detected in HCT116 or U2OS cells before drug

To obtain accurate single-cell intermitotic times (IMTs), apoptosis times (ATs) and cell lineages, we obtained datasets from two previous cell-line studies on colon cancer HCT116 [18] (Fig 1a–1c and Table 1) and osteosarcoma U2OS [37] (Fig 1d–1f and Table 2). These studies performed five days of single-cell time lapse microscopy, tracking cells in two days of drug-free medium followed by three days of cisplatin treatment, with snapshots taken every 30 mins. While HCT116 were exposed to a single concentration of cisplatin approximately near its IC50 value [18], U2OS cells were treated with three concentrations of cisplatin – 7, 10 and 13 μM corresponding to Low, Medium and High labels [37]. The HCT116 dataset had both the daughter cells of dividing cells tracked (Fig 1a) allowing extraction of single-cell times and fates for all the cells in the population as well as lineage relationships between the single cells. The U2OS dataset however was generated by tracking a single randomly chosen daughter after each cell division (Fig 1d). Besides measurements of IMT and AT of single cells, these datasets allow simultaneous extraction of the population dynamics by counting the number of surviving cells in each time frame (Fig 1b and 1e). Detailed statistics for these two cell types can be found in Tables 1 and 2. In both datasets there were a small fraction of cells which never divided through the entire period they were tracked (‘never-dividers’; see Tables 1 and 2). Among these never-dividers in the HCT116 dataset, a higher fraction (10/13) died post cisplatin while the rest survived, reflecting the higher fraction of cell death in the overall population (Table 1). In the U2OS dataset, 2/9, 8/13 and 3/9 never-dividers died post low, medium and high cisplatin respectively (Table 2). Overall, since these never-dividers did not show any clear bias towards any particular end fate (survival or death post cisplatin), and their numbers were low, we excluded these cells from all further analysis. We also excluded a very small fraction of cells that die before cisplatin treatment in the U2OS dataset (Table 2).

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Fig 1. Pre-cisplatin heterogeneity in inter-mitotic times arises from a single population of cycling cells.

(a) Schematic of a single cell lineage from the HCT116 experiment, where both pairs of sisters were tracked after cell division. The dashed vertical line shows time of administration of cisplatin. Blue lines track back to ancestors of surviving (persister) cells while the red lines track back to ancestors of cells that die (sensitive cells). (b) log of population versus time for HCT116 cells shows a distinct bi-phasic exponential decay after cisplatin (denoted by the vertical line). (c) Boxplot comparing IMTs of distinct drug naive ancestor cells belonging to lineages of persister or sensitive phenotypes. (d) Schematic of the U2OS data where tracking records only one randomly chosen daughter cell after a division event (forward lineages). (e) log of population versus time for U2OS cells with three different cisplatin concentrations. The biphasic decay observed in panel (b) is not observed here. (f) Pre-cisplatin single-cell IMT distribution is better explained with a single EMG (Exponentially Modified Gaussian) model than a mixture of 2 EMGs as verified by comparing AIC values for the two models. The histogram represents the IMT distribution obtained from the single-cell microscopy experiment while the lines are the EMG fits.

https://doi.org/10.1371/journal.pcbi.1013446.g001

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Table 1. A summary of the HCT116 dataset.

Detailed reasons for exclusion of the 18 lineages are provided in Table A in S1 Text. 13 out of these 18 are cells that are not observed to divide over the full course of their observed trajectories. 65 single cell lineages proliferate to generate 275 cells at time of cisplatin administration. Out of these 275, 123 die while 52 divide with both daughters ending up dying, resulting in about 64% cell death.

https://doi.org/10.1371/journal.pcbi.1013446.t001

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Table 2. A summary of the U2OS dataset.

The reason for excluding a small fraction of the cells was because they either died pre cisplatin administration, or they never divided.

https://doi.org/10.1371/journal.pcbi.1013446.t002

We first asked whether distinct sub-populations of slowly dividing cells could be detected in the drug naive HCT116 and U2OS cells (after elimination of the never-dividers). To find the population-time curve of the HCT116 cells (Fig 1b), we summed the number of cells in each time frame. The semi-log plot of the curve shows characteristic exponential growth before the addition of the drug (vertical line in Fig 1b) and a bi-phasic exponential decay after cisplatin exposure. The first larger decay rate switches to a plateau after the 100th hour of the experiment, driven by the drug tolerant persisters in the population. We identified persisters as those cells surviving till the end of the experiment (resulting in the second phase of exponential population decay) and extracted the IMTs of their drug-naive ancestors by tracking back in time along their lineages (Fig 1a, blue line). We compared these drug-naive ancestor IMTs to those of the sensitive cells that died after cisplatin treatment (Fig 1a, red line), and could not detect any statistically significant difference between these two distributions (Fig 1c; see Sect 3 in S1 Text for details).

For the U2OS population-time curves, calculated by averaging over the number of cells across 51 fields of view per time frame, showed a characteristic exponential growth before cisplatin addition and concentration-dependent exponential decay post cisplatin exposure (Fig 1e). However, five days of tracking was not sufficient to observe the second slower exponential decay unlike in the HCT116 cells. While this lack of a second slower exponential decay may appear to be limiting for further analyses, this turns out not to be the case. As explained in the subsequent sections, predicting the first exponential decay is the most important mathematical task in this work, which allows developing new models of cell fate. Further, to investigate whether multiple sub-populations with distinct IMT distributions exist pre-cisplatin, we fit the pre-drug IMT distribution with a mixture model (Sect 3 in S1 Text) instead of tracing the ancestors of persisters, thereby circumventing the need for a second exponential phase. We thus compared the ability of single and two-component Exponentially Modified Gaussian (EMG) models to explain the IMT distribution of drug naive cells, using the Akaike Information Criterion (AIC). We found that the single EMG (AIC = 3687.4) fared better than the mixture model (AIC = 4137.649). Also, the two models generated best fit distributions that were essentially indistinguishable, suggesting the absence of multiple sub-populations with distinct proliferation rates within the U2OS population as well (Fig 1f).

Our analysis demonstrates that in both HCT116 and U2OS cell types, after setting aside the never-dividers, the heterogeneity in cell cycle times before addition of cisplatin is best explained by a single population of cycling cells, not by multiple sub-populations with distinctly different cycling kinetics. This suggests that if a pre-DTP state exists amongst these cells before drug administration, it does not have any fitness cost as compared to the sensitive cell state.

Proliferation rate of cancer cells before drug can be predicted from single-cell division times

Having established that a single IMT distribution best explains the datasets before cisplatin treatment, we next asked how accurately this IMT distribution can predict the population growth rates. For exponentially proliferating cells, the population growth rate is approximately given by the inverse of the mean of the IMT distribution. While this is the most popular model used to describe exponential cellular growth, the underlying assumption here is that the IMT distribution is exponential. However, this is clearly not true in our datasets (Fig 1f) and hence a more general approach is required. An age-structured population model (Model M0; see Methods and Sect 9 in S1 Text for details) is a generalization of the simple exponential growth models, that allows for non-exponential IMT distributions, and connects the population growth rate to the full IMT distribution via the Euler-Lotka equation (see Methods and Sect 4 in S1 Text for more details) [38,39,41,43]. From the measured single-cell IMTs we estimated the population growth rates from both the inverse of the mean as well as the Euler-Lotka equation. We compared these predictions with the true growth rate, defined as the linear regression slope of the semi log plot of number of cells as a function of time (Table 3). In both HCT116 and U2OS datasets, there were only about 3% or less cells that died before cisplatin treatment (Tables 1 and 2), hence only the IMT distributions were used in predicting the population growth rates before drug.

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Table 3. An age structured model (Euler-Lotka equation) accurately predicts population growth rates of both HCT116 and U2OS cells before cisplatin treatment.

Shown here are deviations (in %) of growth rates predicted by either the Euler-Lotka equation or the inverse of the mean IMT, from the experimentally measured growth rate (obtained via linear regression of the cell-population versus time curve in Fig 2a and 2d). The growth rate prediction by the Euler-Lotka equation for HCT116 cells is within ∼11% of the error expected from age-structured population simulations with the inferred starting age-distributions. For the U2OS cells, the predictions are off by about ∼ 20%.

https://doi.org/10.1371/journal.pcbi.1013446.t003

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Fig 2. An age-structured model accurately predicts population growth rates from single-cell IMT distributions.

(a)-(c) are plots for HCT116, (d)-(f) are for U2OS. (a) Semi-log plot of population versus time curve of HCT116 cells before cisplatin administration. Predicted growth rate using inverse of the mean (pink line; error = 62%) and the Euler-Lotka equation (blue line; error = 13%) are compared with the experimental data shown as black dots. (b) Maximum Likelihood inference of the age distribution of ancestor cells immediately after seeding. (c) Distribution of errors in Euler-Lotka predictions of growth rate, obtained by running 500 iterations of an age-structured population simulation with initial age distribution taken from panel (b). Vertical dashed line represents the Euler-Lotka prediction error in experimental data (corresponding to error of the blue line in panel (a)). (d) Semi-log plot of population versus time curve of U2OS cells before cisplatin, with colours as in panel (a). Predicted growth rate using inverse of the mean (pink line; error = 73%) and the Euler-Lotka equation (blue line; error = 22%) are compared to the experimental data shown in black dots. (e) Inferred age distribution of seeded population of U2OS cells, as in panel (b). (f) Distribution of errors in Euler-Lotka estimates, obtained by running 500 iterations of simulations as in panel (c). Initial age distribution was taken from panel (e) for these simulations. Vertical dashed line represents Euler-Lotka prediction error in experimental data (corresponding to error of the blue line in panel (d)).

https://doi.org/10.1371/journal.pcbi.1013446.g002

For the HCT116 cells, we found that the inverse of the mean gave an error of around 62% (percentage difference between estimated versus true growth rates), while the Euler-Lotka estimate gave a much reduced error of 13% (Table 3 and Fig 2a). The large error in the former model was not surprising as the measured IMT distribution was neither exponential nor very narrow to be effectively represented by the mean. For the latter result, we further investigated the potential origins of the 13% error. To check whether this error could arise simply due to the small number of cells in our experiments, we performed stochastic simulations taking the measured IMT distribution as input to generate in silico proliferation trajectories with similar cell numbers.

For each simulation run we calculated the growth rate prediction errors using both the growth rate estimates, mimicking the procedure we carried out on the experimental data. Since the age distribution of the cells at the initial time is an input to the simulations, and it is known that this distribution can affect the emergent population dynamics [39], we estimated the age distribution of the initial cells directly from the experimental data (Fig 2b). Based on the IMT distribution of the full pre-cisplatin population, we inferred the starting ages of the initially plated cells in the experiment using a Maximum Likelihood approach (Fig 2b; details in Sect 6 in S1 Text). We then repeated the simulations using the inferred initial age distribution and measured IMT distributions and found that the Euler-Lotka errors peaked around 2% (Fig 2c). We carried out a similar analysis for the U2OS dataset, where we once again found a large error of around 73% from the inverse of the mean prediction, and a much reduced error of 23% from the Euler-Lotka equation (Fig 2d and Table 3). We found that the inferred initial age distribution of the starting U2OS cells was broader than that of the HCT116 cells (Fig 2e), and when used in the simulations, could account for about 4% error in the growth rate prediction (Fig 2f). The unexplained errors of about 10  20% for HCT116 and U2OS cells could be due to artefacts of the image analysis pipeline, or from other unaccounted biological sources such as cell-size control of the cell cycle which might induce lineage correlations in cell cycle times and consequently modifications to the growth rate [44].

Our results in this section demonstrate that in the absence of cisplatin, the age-structured model predictions of population growth rate using a single IMT distribution is accurate to within 10  20% in both HCT116 and U2OS cell types. This provides a benchmark for the expected error level when trying to predict population dynamics from our single-cell datasets. Additionally, the fact that a single IMT distribution is sufficient to predict the population growth rate is consistent with results from the previous section, which suggested the absence of sub-populations of cells with different cycling kinetics.

First exponential decay of drug treated cancer cells is not established by single-cell division and death times

Since the age-structured model (Model M0) could predict pre-cisplatin population growth rates from the measured cell division times within 20% error, we next asked if a similar prediction accuracy can be achieved in the post-cisplatin scenario, where an added dimension is cell death causing the populations to decay instead of grow. Throughout this section, we will be focusing only on the first phase of exponential decay after cisplatin treatment (for the U2OS cells the experiments were not long enough to observe the second exponential phase). To find the IMT and AT distributions for both HCT116 and U2OS cells (Fig 3a–3d), we extracted the times from cells that were either born before drug exposure but divided or died after drug exposure, or cells that were both born and reached a fate after drug exposure. The primary reason for having to include cells born before drug administration was because there were very few cells born after cisplatin exposure that went on to finish a full cell cycle to divide again. There were no HCT116 cells that finished a full division event purely post cisplatin (Table 1), while in the U2OS datasets there were just 2/232, 2/240 and 1/296 such cells in the Low, Medium and High concentrations respectively (Table 2). A careful deconvolution of the pre versus post drug cell cycle times from these ‘straddling cells’ therefore needs to be performed via inference [18], as we describe later in this section.

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Fig 3. Inferring the underlying post-drug IMT and AT distributions from the experimentally measured, biased distributions.

(a)-(d) are plots of measured IMT and AT distributions of HCT116 and U2OS cells while (e)-(h) are the competing-risks corrected distributions inferred using a Markov Chain Monte Carlo approach (see Methods and Sect 8 in S1 Text for details). (a)-(b) IMT and AT distributions of HCT116 cells. All cells that divided or died during the treatment period were tracked and measured, including cells that were born before and after cisplatin addition. (c)-(d) Forward lineage distributions for U2OS cells that have division and death events that occurred during the treatment period. The three distributions in both panels (c) and (d) are almost identical, demonstrating that increasing drug concentration does not affect division or death rates post cisplatin. For the three IMT distributions, Kruskal-Wallis test gave a p-value of 0.22 while for the three AT distributions, Kruskal Wallis test gave a p-value of 0.53. (e) Inferred IMT distribution of HCT116 cells. Compared to panel (a), the long tail is evident demonstrating the significant bias in the measured distribution. (f) Inferred AT distribution of HCT116 cells, which is similar to the measured distribution in panel (b). (g) Inferred IMT distributions for U2OS cells reveal increasingly long tails for increasing cisplatin concentrations. (h) Similar to panel (f), the inferred AT distributions for U2OS cells is almost identical to the measured AT distributions.

https://doi.org/10.1371/journal.pcbi.1013446.g003

Though the U2OS decay rates were distinctly different between High (), Medium () and Low () cisplatin, which amounts to a 3-fold change between High and Low concentrations, surprisingly there were no statistically significant differences in the measured IMT and AT distributions across the three cisplatin concentrations (Fig 3c and 3d respectively). This directly showed, in a model-independent manner, that times to division or death post cisplatin (and hence the division and death rates) do not determine the population decay rates. This result contradicts the widely used assumption that the effect of drugs can be well described either as increasing the cellular death rate or reducing the division rate [31–36].

To rigorously and quantitatively confirm that the minor differences in the measured IMT and AT distributions cannot explain the large differences in decay rates across cisplatin concentrations, we calculated the predicted rates using both inverse of the mean and Model M0 (see Sect 9 in S1 Text for details). The IMT and AT distributions were not sufficient to accurately predict the concentration-dependent decay rates of the U2OS cells, producing errors greater than 100% (‘Measured’ columns in Table 4). Indeed, even for HCT116 cells where we had data from one drug concentration, the error in predicted population decay rates was greater than 100% (‘Measured’ column in Table 4). Taken together, these findings were rather surprising in light of our previous section results as well as the vast body of literature modeling population growth/decay as the difference in birth and death rates.

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Table 4. Age structured models fail to predict population decay rates for cells treated with cisplatin.

Shown here are deviations (in %) of the predicted decay rates from the measured decay rates (obtained via linear regression of post-cisplatin experimental data in Fig 1b and 1e). Predictions were made using IMT and AT distributions obtained directly from the experiment (‘Measured’) and from competing risks-corrected unbiased distributions (‘Inferred’).

https://doi.org/10.1371/journal.pcbi.1013446.t004

We therefore wondered if our analysis using the measured IMT and AT distributions suffered from a competing-risks bias, a statistical effect that prevents direct measurement of the correct underlying distributions when the events are mutually exclusive. We have previously demonstrated that this effect leads to strong biases in the IMT and AT distributions measured after drug treatment [18], and hence can potentially lead to erroneous population growth rate predictions. A simple analytic calculation for exponentially distributed IMT and AT is presented in Sect 8 in S1 Text that provides intuition for this competing risks effect, and how it can be corrected using inference techniques. In the more complex case of non-exponentially distributed IMT and AT as is the case with the HCT116 and U2OS datasets, we used a Markov Chain Monte Carlo (MCMC) approach to infer the parameters of the unobserved distributions (see Sect 8 in S1 Text for details). The inferred IMT distributions of HCT116 cells (Fig 3e) showed a longer right tail (slower division times) compared to the measured IMT distribution, which was not observed for the AT distribution (Fig 3f). For the U2OS cells, the right tails of the inferred IMT distributions increased with higher drug concentrations (Fig 3g) while once again the inferred AT distributions (Fig 3h) had similar right tails compared to the measured AT. Using these inferred IMT and AT distributions we then recalculated the predicted decay rates using the mean inverse and the Eula-Lotka equation. While the errors significantly reduced (‘Inferred’ columns in Table 4), they were still very high with the smallest error around 40% in both HCT116 and U2OS and the largest error greater than 100% for low and medium cisplatin doses in U2OS.

In summary, we found that estimates of population decay rate using measured single-cell times to division and death post cisplatin treatment were erroneous by over 100%, predicting positive growth instead of decay. Even after correcting for competing risk biases, the population decay rates could be predicted at best with an error of 40%, often with errors as high as 100%. These counter-intuitive results suggest that measured single-cell IMT and AT are not sufficient to estimate population decay rates post addition of cisplatin. In turn these results suggest that post-drug stochastic competition of cell fates inherent to these exponential growth models (Model M0), is likely to be an incorrect model of cell-fate decisions in the development of cancer drug persistence.

Cell-state inheritance with fate decisions before and during drug treatment control cancer population dynamics

We next asked if the nature and timing of fate decisions can be determined by developing a series of models with varying assumptions and checking for compatibility with experimental data. The large errors in Model M0’s predictions of decay rates suggest that the cell fates are likely decided either during or before drug addition, largely independent of the single-cell IMT and AT distributions. Therefore we developed three alternate models of the fate decision process, Models M1, M2 and M3 (Fig 4). In these models, cell fates are either (i) determined based on 2 cell-states before drug – a pre-DTP state P’ that results in surviving persisters post drug (some of which can divide), and a sensitive state S’ that leads to cell death after drug administration (Model M1), (ii) decided randomly for each cell at the time of drug addition with probabilities that depend on the drug concentration (Model M2), or (iii) a combination of both M1 and M2 where pre-existing cell-states determine the end fate with large probability while the drug concentration modulates this probability to some extent (Model M3). M1 therefore describes a purely pre-drug fate decision scenario while M2 describes purely drug-induced fate decisions, effectively modeling a scenario where persister cells emerge exclusively after drug treatment, in a concentration dependent manner.

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Fig 4. Schematics of models M0 to M3, highlighting the differences in timing and mode of cell fate determination of each model.

The models are shown with the possible state and state-to-fate transitions on top, along with an example lineage below. Cells in gray indicate no role of cell-states in determination of fate. Coloured cells indicate pre-existing cell-states that pre-determine final fates to a large extent. The dice indicates a probabilistic state-to-fate map, that assigns fates stochastically to each single cell, conditioned on its state at the time of drug administration. A syringe accompanying the dice indicates drug concentration dependent state-to-fate probabilities. The state-to-fate map is modeled with a ‘Fate Matrix’ F. Thicker arrows indicate higher probabilities in the F matrix. ‘ToD’ is Time of Drug and ‘EoE’ is End of Experiment. In M0, fate decisions happen entirely post-drug via hazards of division and death drawn from the post-drug IMT and AT distributions. In models M1-M3 the IMT and AT distributions are essentially irrelevant, except for setting an overall timescale. In M1, fate decisions are based purely on pre-existing cell states. In M2, single cells get assigned their fate at time of drug administration based on drug concentration-dependent probabilities. M3 is a combination of M1 and M2, where fates are largely pre-determined by cell states, but get modulated at time of drug administration to some degree. For example, as depicted in the lineage in Model M3, cells in state S’ at ToD mostly die post drug, but a few survive.

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M3 combines elements of both models M1 and M2, and captures the existence of cell states before drug that predispose cells towards survival or persistence post drug. Inter-conversion between states S’ and P’ in Models M1 and M3 is modeled using a Markov Chain described by a transition matrix T while the drug concentration-dependent fate probabilities in Model M3 are imposed using a fate matrix F (Fig 4). In effect, F models cell-state switching post drug treatment, by allowing a small fraction of pre-drug S’ cells to persist and P’ cells to die post drug treatment. Further details and parameterizations of each model are provided in Methods and Sects 9 and 10 in S1 Text. We found that Models M1 and M2 could recapitulate only certain aspects of the experimental datasets (see S1 for details). M1 generated clear biphasic exponential decays and could explain the lineage correlations, but not the concentration-dependence of the first exponential decay. This is because every cell is already fated to either die, or survive/divide based on its cell state at time of drug addition. The drug itself plays no role in determining fate. M2 on the other hand could generate biphasic exponential decays that were drug concentration dependent, but could not generate lineage correlations due to the lack of cell-states. Only Model M3 could quantitatively explain all three major observations simultaneously: (1) decay rate for a fixed drug concentration, (2) lineage correlations in end-fate (Fig 5a) and (3) drug concentration dependence of the population decay rates. On simulating M3, where the pre-drug transitions create a steady-state distribution of S’ and P’ cells (Fig 5b), we noticed that a broad spectrum of effective decay rates could be obtained after cisplatin exposure even with the IMT and AT distributions fixed. This arises either due to different Fate matrices for a given Transition matrix (Fig 5c) or due to different Transition matrices for a given Fate matrix (Fig 5d). This explains the surprising results from Fig 3c–3d where the IMT and AT distributions were found to be insensitive to drug concentration – the IMT and AT distributions have little contribution towards establishing the decay rate. Rather, the population dynamics is dominated by the relative proportion of the two states at time of drug exposure (determined by T), and the entries of F. Finally, by performing a coarse parameter sweep we found that for certain physically realistic values of the T and F matrices, we could approximately recapitulate the experimentally measured decay rates (Fig 5e) as well as the lineage correlations (Fig 5f) while maintaining the same measured IMT and AT distributions as observed in the HCT116 experiment.

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Fig 5. A two state Markov model with state-dependent fates (Model M3) quantitatively explains the experimentally measured decay rate and correlations in lineage related HCT116 cells.

(a) Lineage correlations in fates of HCT116 cells measured from time lapse experiments. Correlations exist for first and second cousins (FC and SC), while third cousins (TC) are similar to randomly selected pairs of non-lineage related cells (horizontal dashed line). (b) An example run of Model M3 showing the relative populations of the two states as a function of time before drug administration. (c) Population versus time trajectories from simulations of Model M3 using identical IMT and AT distributions, a fixed Transition matrix T, but different Fate matrices. Each colour corresponds to simulations with a different F. (d) Same as panel (c), but with different Transition matrices for a fixed Fate matrix F. (e) Distribution of decay rates from the simulations (red histogram). The vertical dashed line is the decay rate obtained from the HCT116 experimental dataset, . (f) Comparison of lineage correlations obtained from the HCT116 experiment (red) with simulations of Model M3 (blue). Transition rates used in the simulation were symmetric for transition between the two states, per hour. The fate probabilities for cells to die, divide and stay alive for the two states were: Sensitive state S’ = (0.9125, 0.05, 0.0375); pre-DTP state P’ = (0.35, 0.1, 0.55).

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The fate probabilities for cells to die, divide and stay alive for the two states were: Sensitive state S’ = (0.9125, 0.05, 0.0375); pre-DTP state P’ = (0.35, 0.1, 0.55). These numbers support our expectation that though cells in the sensitive state are more likely to die (with probability 0.91) and pre-DTP state more likely to survive (probability 0.55), with some small but finite probability S’ cells survive (probability 0.037) and P’ cells die (probability 0.35). Note however that these numbers should be treated only as approximate estimates, with further work required to develop accurate parameter inference from these datasets.

Our analysis in this section quantitatively explains why the birth and death rates (and more generally the single-cell IMT, AT distributions) do not determine the decay rates of drug treated cells at the population level. We demonstrate that the transition rates between cell states before drug treatment, the ability of these states to be passed down across cellular generations, and the cisplatin concentration-dependent probabilities of these states to result in death or survival largely determine the population dynamics and lineage correlations of cells after drug treatment.

Change in barcode diversity before and after drug cannot establish timing of fate decisions in the presence of cell-state switching

Our analyses thus far demonstrated the importance of pre-existing cellular states in the population that leads to persistence of cells in the face of cisplatin treatment. This conclusion was true for both colorectal cancer as well as osteosarcoma cell types. However, a recent study on colorectal cancer cells concluded that the persister fate arose after the addition of cisplatin, based upon analysis of a high-complexity barcode library that had been transduced into the cancer cells [25]. In short, this study found that the barcode diversity (measured by the Shannon Diversity Index) showed no reduction upon cisplatin treatment, which suggested that the persister fate choice was likely made post drug treatment. Since these results are conceptually different from our findings, we next investigated potential reasons that may explain the apparently contradictory results.

The barcoding approach to determine the timing of persister fate choice relies on comparing the barcode diversity, or abundance of unique barcodes, before and after drug exposure in many cellular replicates. A reduction in barcode diversity after drug treatment is interpreted as selection of a small set of pre-existing persister cells (Fig 5a i). Maintenance of barcode diversity on the other hand suggests a scenario where persister cells arise randomly in cells after drug treatment, thereby leading to many surviving barcodes (Fig 5a ii). However, these interpretations were originally derived in the context of genetic mutations [45] where reversal of mutations is rare. The same assumptions do not hold in case of non-genetic cell states (the pre-DTP state or sensitive state in our case), and we argued in the previous sections that transitions between the two states is a fundamental aspect of the dynamics in both HCT116 and U2OS cell types.

To quantitatively explore how cell-state switching before drug addition could affect interpretations of barcode diversity, we used Model M3 with the same parameters from the previous section, with the additional information of barcodes (Fig 6a iii–iv; see Sects 11 and 12 in S1 Text for details). In brief, each starting cell was given a barcode ID that was inherited by all descendants across cell divisions and maintained over cell-state changes. The fates were conditioned on cell-state as described before for Model M3. The results of the simulation are shown as an abundance bubble plot (Fig 6b) for qualitative visualisation and as a Shannon Diversity Index (SDI) plot (Fig 6c) for quantitative assessment of barcode enrichment or diversity. The drop in Shannon Diversity Index post drug was observed only when the simulation did not allow for state transitions before drug treatment, which is equivalent to very slow transitions compared to a typical cell cycle time (Fig 6b–6c, ‘No Trans’). However, if the transitions were allowed with rates that gave rise to the experimentally measured lineage correlations (Fig 5f), it resulted in no change in the SDI value (Fig 6b–6c, ‘SimExpt’). This short analysis confirmed that the timing of development of pre-DTP states cannot be inferred using current barcode analysis techniques and might require development of novel methods to account for cell-state transitions.

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Fig 6. Maintenance of barcode diversity before and after drug cannot be interpreted as a signature of drug-induced cell fates.

(a) Current barcoding methods relate fate determination to pre-drug cell-states if the barcode diversity, shown here by the presence of distinct barcodes (colored circles), significantly reduces after drug (i). However, if the diversity does not reduce then fate determination is assumed to be in response to drug exposure (ii). Our model however, suggests that both the conditions (i & ii) can be obtained with fates largely determined before drug administration. Simulation results show that in conditions when the cell-states before drug transition on time scales much slower than the cell division time, barcode diversity shrinks significantly post drug treatment (iii). However, if the transition rates are of the order of a few cell cycles or faster, more sensitive cells transition to a pre-DTP state, resulting in insignificant change in barcode diversity post-drug (iv). (b) Barcode abundance plot post drug treatment obtained from simulations of Model M3. Circle fill colors represent distinct barcodes while their sizes show barcode frequency counts. Results of 3 simulations are compared. ‘Control’: simulations with cell-state independent fate assignment of cells; ‘No Trans’: simulations with cell-fate assignment conditional on cell-states that do not inter-convert; ‘SimExpt’: simulations with state dependent fate assignment with cell-state transitions allowed before drug addition. (c) Diversity of barcodes can be measured by the Shannon Diversity Index. The boxplot here shows change in Shannon Diversity Index for the corresponding simulations described in panel (b). (d) Distribution of persister count fractions across lineages in the HCT116 dataset and Model M3 simulations. (e) Variance over Mean Ratio (VMR) of persister cell counts as measured in the experiment (Expt) and the Model M3 simulation matching experimental results (Sim). The horizontal dashed line shows the VMR for a Binomial distribution with p = 0.36

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Finally, we used an orthogonal approach to analyse the timing of fate decisions for the HCT116 cells – a Luria-Delbrück (LD) fluctuation test. For U2OS cells this analysis was not possible since the second exponential decay phase was not observed, precluding identification of persisters (Fig 1e). The LD test is based on quantifying the Variance over Mean Ratio (VMR) of the persister counts across lineages post cisplatin. A scenario where the persister fate is assigned with probability p at time of drug administration, and the persisters do not proliferate post drug, would lead to a Binomial distribution in the number of persisters and a consequent VMR given by 1–p. On the other hand, if the persister fates were pre-determined based on pre-drug inheritable cell-states, then the VMR would be greater than 1 [46]. We therefore calculated the VMR from both the HCT116 experimental dataset as well as our simulations of Model M3. To ensure that cell divisions post drug did not bias our calculation of the VMR, we identified all extant cells at the time of drug administration and tracked their final post-drug fates, randomly choosing only one of the two daughters in cases where the cell divided post drug (see Methods and Sect 13 in S1 Text for more details). Using this method, we first checked that the ModelM3 generated similar fractions of persister cells across lineages as in the HCT116 data (Fig 6d). Next, we computed the VMRs: for the HCT116 data, the Binomial distribution predicts a VMR of 0.64, since the fraction of persisters was approximately p = 0.36 (see Table 1; out of 275 cells at time of cisplatin, 123 cells died while 52 divided and then died). However, the calculated VMR was around 2.4 (Fig 6e; see Sect 13 in S1 Text for details). Model M3 simulations with parameters that reproduced experimentally observed decay rates (Fig 5e) and lineage correlations (Fig 5f), generated a VMR of 1.6 (Fig 6e). Therefore both the experimental data and simulations suggest that end fates could not have been assigned solely as a post-drug phenomenon, consistent with the results of our analysis using lineage correlations. In summary, the results in this section show that change in barcode diversity before and after drug treatment is not always sufficient to establish the timing of fate decisions. This is true when fates are related to non-genetic cell-states which inter-convert between each other on time scales similar to the cell-cycle time. Maintenance of barcode diversity after drug treatment is therefore consistent with a pre-DTP state existing before drug treatment and driving fate outcomes, as our analyses demonstrates using both lineage correlations as well as the Luria-Delbrück fluctuation test.