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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.

No MeSH data available.


Related in: MedlinePlus

Simulation results corresponding to the competition between the sub-populations of a heterogeneous populations. We show how the average extinction time of either population, TE, varies as the carrying capacity, K, changes. We observe that the dependence is linear. For simplicity, the two sub-populations are assumed to be identical (i.e. with the same characteristic parameter values for both the intracellular dynamics (cell-cycle) and the population-level dynamics (birth and death rates)). The carrying capacity  by varying the death rates of both populations. The values of the death rates are , which correspond to , respectively. Averages are done over 500 realisations of the hybrid stochastic model.
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f0055: Simulation results corresponding to the competition between the sub-populations of a heterogeneous populations. We show how the average extinction time of either population, TE, varies as the carrying capacity, K, changes. We observe that the dependence is linear. For simplicity, the two sub-populations are assumed to be identical (i.e. with the same characteristic parameter values for both the intracellular dynamics (cell-cycle) and the population-level dynamics (birth and death rates)). The carrying capacity by varying the death rates of both populations. The values of the death rates are , which correspond to , respectively. Averages are done over 500 realisations of the hybrid stochastic model.

Mentions: In order to numerically check this scenario, we could proceed to estimate the survival probability at time t, PS(t). However, since this quantity exhibits a fat-tail behaviour, this would be computationally costly. A more efficient method is to resort to the asymptotic of the extinction time with system size, which in this case can be identified with the carrying capacity, K (Hidalgo et al., 2015). Typically, a quasi-neutral competition is associated with an algebraic dependence of the average extinction time of either population, TE, on the system size, in this case determined by the carrying capacity, KDoering et al., 2005, Kogan et al., 2014, Lin et al., 2012. In Fig. 11 we plot simulation results for the competition between two identical populations. In particular, we study how the average extinction time of either population, TE, varies as the carrying capacity is changed. We observe that this quantity exhibits a linear dependence on the carrying capacity:(37)TE∼K,


Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis
Simulation results corresponding to the competition between the sub-populations of a heterogeneous populations. We show how the average extinction time of either population, TE, varies as the carrying capacity, K, changes. We observe that the dependence is linear. For simplicity, the two sub-populations are assumed to be identical (i.e. with the same characteristic parameter values for both the intracellular dynamics (cell-cycle) and the population-level dynamics (birth and death rates)). The carrying capacity  by varying the death rates of both populations. The values of the death rates are , which correspond to , respectively. Averages are done over 500 realisations of the hybrid stochastic model.
© Copyright Policy - CC BY
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5016039&req=5

f0055: Simulation results corresponding to the competition between the sub-populations of a heterogeneous populations. We show how the average extinction time of either population, TE, varies as the carrying capacity, K, changes. We observe that the dependence is linear. For simplicity, the two sub-populations are assumed to be identical (i.e. with the same characteristic parameter values for both the intracellular dynamics (cell-cycle) and the population-level dynamics (birth and death rates)). The carrying capacity by varying the death rates of both populations. The values of the death rates are , which correspond to , respectively. Averages are done over 500 realisations of the hybrid stochastic model.
Mentions: In order to numerically check this scenario, we could proceed to estimate the survival probability at time t, PS(t). However, since this quantity exhibits a fat-tail behaviour, this would be computationally costly. A more efficient method is to resort to the asymptotic of the extinction time with system size, which in this case can be identified with the carrying capacity, K (Hidalgo et al., 2015). Typically, a quasi-neutral competition is associated with an algebraic dependence of the average extinction time of either population, TE, on the system size, in this case determined by the carrying capacity, KDoering et al., 2005, Kogan et al., 2014, Lin et al., 2012. In Fig. 11 we plot simulation results for the competition between two identical populations. In particular, we study how the average extinction time of either population, TE, varies as the carrying capacity is changed. We observe that this quantity exhibits a linear dependence on the carrying capacity:(37)TE∼K,

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