Quantifying replicative senescence as a tumor suppressor pathway and a target for cancer therapy

Abstract

To study quantitatively replicative senescence as a tumor suppressor mechanism, we investigate the distribution of a growing clonal cell population restricted by Hayflick's limit. We find that in the biologically relevant range of parameters, if the imbalance between cell division and death is moderate or low (high death-to-birth ratio), senescence offers significant protection against cancer by halting abnormal cell proliferation at early pre-diagnostic stages of tumor development. We also find that by the time tumors are typically detected, there is a high probability that telomerase is activated, even if the cell of origin was telomerase negative. Hence, the fact that most cancers are positive for telomerase is not necessarily an indication that cancer originated in a telomerase positive cell. Finally, we discuss how the population dynamics of cells can determine the outcomes of anti-telomerase cancer therapies, and provide guidelines on how the model could potentially be applied to develop clinically useful tools to predict the response to treatment by telomerase inhibitors in individual patients.

Figure 1

(A) Cells have a limited replication capacity equal to the maximum number of times they may divide. A cell with replication capacity ρ > 0 can either divide with probability q or die with probability 1 − q. If division occurs the replication capacity of each daughter cell decreases by one. (B) Expected number of cells vs time. At time t = 0 there is one single cell with replication capacity ρ = 30 and division probability q = 0.7. The solid line is calculated using Eq. 7. Dots are the average of 1000 independent random simulations. (C,D) Simulations of the total number of cells (C) and the cumulative number of divisions (D) calculated using the hybrid algorithm (solid lines) and the fully stochastic algorithm (dots). In the hybrid algorithm the switch between stochastic and deterministic phases takes place when the number of proliferating cells equals 10⁶. Panel (C) plots the trajectories after the stochastic-deterministic transition takes place (x-axis indicates time elapsed after the transition). Panel (D) plots the whole span of the simulations. The parameters (k = 40, q = 0.75) are such that the implementation of the fully stochastic algorithm through the entire lifetime of the population is computationally feasible. See text for discussion.

Figure 2

(A)Maximum size of the expected cell population as a function of the replication capacity (k) and the death-to-birth ratio (r) of the originating cell. Dashed lines indicate the regions where the maximum population is greater than 10⁹ cells (detection threshold) and 10¹² cells (lethal burden). See text for discussion. (B) Region A identifies parameters for which the expected maximum population size is greater than 10⁶. (C) Heat map: Probability that the maximum population size is greater than 10⁹. Dashed line: region where the maximum size of the expected cell population, , is greater than 10⁹. (D) Heat map of the probability of escaping replication limits as a function of the replication capacity of the originating cell and the death-to-birth ratio. The value of P(erl) is estimated as the midpoint between the upper and lower bounds in Eq. 9. Results based on 10⁴ simulations per point.

Figure 3. In panels (A,B) the distributions of the maximum population size are bimodal.

In (A) the maximum number of cells was either more than 2.2 × 10⁷ (major axes, 75% of the time) or less than 5 (inset, 25% of the time); parameters k = 45 and r = 0.25. In (B), which uses parameters k = 45 and r = 0.45, 99.5% of the time the maximum number of cells was either more than 50,000 (major axes, 53.8%) or less than 8 (inset, 45.7%). In (C) the distribution is unimodal (a decreasing function of the maximum number of cells); parameters k = 21 and r = 0.75. In (C) the graph is broken down in two parts (major axes and inset) to present a greater resolution. The frequency of times that the maximum number of cells equaled 12 (last value in the inset) is more than the frequency of times that equaled 13 (first value in the major axes). (D) Coefficient of variation for samples in which the cell population reaches a large size. Number of simulations: 10⁴ in panels (A,B,D); 10⁷ in panel (C).

Figure 4

(A) Probabilities of reaching a large population (depicted by symbols) as a function of the death-to-birth ratio (r) for several values of the replication capacity k. A solid line plots the approximation (1 − r). (B) Probability that the maximum number of cells is greater than 10⁹ (for most values of r the probability of reaching 10⁹ cells is either zero or very close to the approximate probability of reaching a large size (1 − r)). (C) Probability that at least one mutation takes place P(M > 0). (D) Probability of escaping replication limits through a mutation that activates telomerase P(erl). For each of the values of k, P(erl) falls inside the corresponding grey band. The lines that form the boundary of the bands are calculated using the upper and lower bounds in Eq. 9. In (C,D) the mutation rate per cell division μ = 10⁻⁹. Results based on 10⁴ simulations per point.