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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.


Plots showing simulation results of the stochastic multi-scale dynamics of a cell population. These plots show how, in agreement with our steady-state analysis, the population evolves until it reaches a steady-state where the population of resident cells fluctuates around its associated carrying capacity Eq. (30). Colour code: blue lines show the total resident cell population at time t, N(t) (panels (a) and (b)). Green lines (panels (c) and (d)) show the associated oxygen concentration, c(t). The results shown in this figure correspond to a single realisation of the process. Parameter values: , , ,  in panels (a) and (c), and  in panels (b) and (d). (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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f0045: Plots showing simulation results of the stochastic multi-scale dynamics of a cell population. These plots show how, in agreement with our steady-state analysis, the population evolves until it reaches a steady-state where the population of resident cells fluctuates around its associated carrying capacity Eq. (30). Colour code: blue lines show the total resident cell population at time t, N(t) (panels (a) and (b)). Green lines (panels (c) and (d)) show the associated oxygen concentration, c(t). The results shown in this figure correspond to a single realisation of the process. Parameter values: , , , in panels (a) and (c), and in panels (b) and (d). (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 verify the age-structured SSA proposed in Section 4.1, we compare its results with the mean-field predictions, which should be in agreement with the stochastic behaviour of the system for large values of the carrying capacity, . Results are shown in Fig. 9. We observe that, as predicted by our steady-state analysis, the stochastic simulations show how the resident population goes through an initial (oxygen-rich) phase of exponential growth. As the population grows, oxygen is depleted and the resident population eventually saturates onto a number of cells which fluctuates around the mean-field prediction of the carrying capacity (Eq. (30)).


Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis
Plots showing simulation results of the stochastic multi-scale dynamics of a cell population. These plots show how, in agreement with our steady-state analysis, the population evolves until it reaches a steady-state where the population of resident cells fluctuates around its associated carrying capacity Eq. (30). Colour code: blue lines show the total resident cell population at time t, N(t) (panels (a) and (b)). Green lines (panels (c) and (d)) show the associated oxygen concentration, c(t). The results shown in this figure correspond to a single realisation of the process. Parameter values: , , ,  in panels (a) and (c), and  in panels (b) and (d). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
© Copyright Policy - CC BY
Related In: Results  -  Collection

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

f0045: Plots showing simulation results of the stochastic multi-scale dynamics of a cell population. These plots show how, in agreement with our steady-state analysis, the population evolves until it reaches a steady-state where the population of resident cells fluctuates around its associated carrying capacity Eq. (30). Colour code: blue lines show the total resident cell population at time t, N(t) (panels (a) and (b)). Green lines (panels (c) and (d)) show the associated oxygen concentration, c(t). The results shown in this figure correspond to a single realisation of the process. Parameter values: , , , in panels (a) and (c), and in panels (b) and (d). (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 verify the age-structured SSA proposed in Section 4.1, we compare its results with the mean-field predictions, which should be in agreement with the stochastic behaviour of the system for large values of the carrying capacity, . Results are shown in Fig. 9. We observe that, as predicted by our steady-state analysis, the stochastic simulations show how the resident population goes through an initial (oxygen-rich) phase of exponential growth. As the population grows, oxygen is depleted and the resident population eventually saturates onto a number of cells which fluctuates around the mean-field prediction of the carrying capacity (Eq. (30)).

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.