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Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis

View Article: PubMed Central - PubMed

ABSTRACT

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.

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Stochastic simulation results showing typical realisations associated with rescue of quiescence cells (plots (a) and (c)) and recovery of the proliferating population (plots (b) and (d)) upon application of a cell-cycle dependent therapy. The efficiency of the therapy is characterised by the survival fraction, FS. Quiescence rescue is achieved when the survival fraction is set to a value which falls below the critical threshold, Eq. (54). If  (plots (a) & (c)), the cell killing triggered by the therapy is enough to re-oxygenate the population above the activation threshold of the quiescent cells. By contrast, if  (plots (b) and (d)), re-oxygenation is not enough to rescue latent cells from quiescence. Parameter values: , , , , . The subindex “1” corresponds to the active population whilst the subindex “2” denotes quantities associated with the quiescent population. The critical oxygen (as defined in 3.5.2, 3.6) is  for the active cells and  for the quiescent cells. Colour code: I all of the panels in this figure, blue (red) lines correspond to the time evolution of the total number of proliferating (quiescent) cells and green lines, to the time evolution of the oxygen concentration. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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f0060: Stochastic simulation results showing typical realisations associated with rescue of quiescence cells (plots (a) and (c)) and recovery of the proliferating population (plots (b) and (d)) upon application of a cell-cycle dependent therapy. The efficiency of the therapy is characterised by the survival fraction, FS. Quiescence rescue is achieved when the survival fraction is set to a value which falls below the critical threshold, Eq. (54). If (plots (a) & (c)), the cell killing triggered by the therapy is enough to re-oxygenate the population above the activation threshold of the quiescent cells. By contrast, if (plots (b) and (d)), re-oxygenation is not enough to rescue latent cells from quiescence. Parameter values: , , , , . The subindex “1” corresponds to the active population whilst the subindex “2” denotes quantities associated with the quiescent population. The critical oxygen (as defined in 3.5.2, 3.6) is for the active cells and for the quiescent cells. Colour code: I all of the panels in this figure, blue (red) lines correspond to the time evolution of the total number of proliferating (quiescent) cells and green lines, to the time evolution of the oxygen concentration. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Mentions: In order to check the accuracy of the mean-field analysis carried out in Section 6.2 regarding the critical survival fraction for rescue from quiescence. We start by showing (Fig. 12) two typical realisations of the stochastic population dynamics which illustrate the rescue mechanism. In these simulations, we first let the active population settle on to its steady state. We then apply a sustained therapy with constant survival fraction. A more aggressive treatment (FS=0.6 in Fig. 12) greatly affects the active population: the amount of active cells killed by the therapy induces re-oxygenation of the population above the critical oxygen level for activation of the quiescent population whereupon the quiescent cells become proliferating. In Figs. 12(a) and (c), we show that upon activation of the quiescent population, a competition between both populations ensues, which eventually leads to extinction of the active population. A less aggressive therapy (FS=0.7 in Fig. 12) also induces death of the active population and re-oxygenation. However, in this case, the latter is not intense enough to induce activation of the quiescent cells (see Fig. 12(d)) and therefore the active cells will repopulate the system as the quiescent population stays on its course to eventual extinction, as shown in Fig. 12(b).


Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis
Stochastic simulation results showing typical realisations associated with rescue of quiescence cells (plots (a) and (c)) and recovery of the proliferating population (plots (b) and (d)) upon application of a cell-cycle dependent therapy. The efficiency of the therapy is characterised by the survival fraction, FS. Quiescence rescue is achieved when the survival fraction is set to a value which falls below the critical threshold, Eq. (54). If  (plots (a) & (c)), the cell killing triggered by the therapy is enough to re-oxygenate the population above the activation threshold of the quiescent cells. By contrast, if  (plots (b) and (d)), re-oxygenation is not enough to rescue latent cells from quiescence. Parameter values: , , , , . The subindex “1” corresponds to the active population whilst the subindex “2” denotes quantities associated with the quiescent population. The critical oxygen (as defined in 3.5.2, 3.6) is  for the active cells and  for the quiescent cells. Colour code: I all of the panels in this figure, blue (red) lines correspond to the time evolution of the total number of proliferating (quiescent) cells and green lines, to the time evolution of the oxygen concentration. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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f0060: Stochastic simulation results showing typical realisations associated with rescue of quiescence cells (plots (a) and (c)) and recovery of the proliferating population (plots (b) and (d)) upon application of a cell-cycle dependent therapy. The efficiency of the therapy is characterised by the survival fraction, FS. Quiescence rescue is achieved when the survival fraction is set to a value which falls below the critical threshold, Eq. (54). If (plots (a) & (c)), the cell killing triggered by the therapy is enough to re-oxygenate the population above the activation threshold of the quiescent cells. By contrast, if (plots (b) and (d)), re-oxygenation is not enough to rescue latent cells from quiescence. Parameter values: , , , , . The subindex “1” corresponds to the active population whilst the subindex “2” denotes quantities associated with the quiescent population. The critical oxygen (as defined in 3.5.2, 3.6) is for the active cells and for the quiescent cells. Colour code: I all of the panels in this figure, blue (red) lines correspond to the time evolution of the total number of proliferating (quiescent) cells and green lines, to the time evolution of the oxygen concentration. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Mentions: In order to check the accuracy of the mean-field analysis carried out in Section 6.2 regarding the critical survival fraction for rescue from quiescence. We start by showing (Fig. 12) two typical realisations of the stochastic population dynamics which illustrate the rescue mechanism. In these simulations, we first let the active population settle on to its steady state. We then apply a sustained therapy with constant survival fraction. A more aggressive treatment (FS=0.6 in Fig. 12) greatly affects the active population: the amount of active cells killed by the therapy induces re-oxygenation of the population above the critical oxygen level for activation of the quiescent population whereupon the quiescent cells become proliferating. In Figs. 12(a) and (c), we show that upon activation of the quiescent population, a competition between both populations ensues, which eventually leads to extinction of the active population. A less aggressive therapy (FS=0.7 in Fig. 12) also induces death of the active population and re-oxygenation. However, in this case, the latter is not intense enough to induce activation of the quiescent cells (see Fig. 12(d)) and therefore the active cells will repopulate the system as the quiescent population stays on its course to eventual extinction, as shown in Fig. 12(b).

View Article: PubMed Central - PubMed

ABSTRACT

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.

No MeSH data available.


Related in: MedlinePlus