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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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Related in: MedlinePlus

Schematic representation of the different elements that compose our multi-scale model. We show the different levels of biological organisation as well as associated characteristic time scales (Guerrero and Alarcón, 2015) associated to each of these layers: resource scale, i.e. oxygen which is supplied at a constant rate and consumed by the cell population, cellular scale, i.e. oxygen-regulated cell cycle progression which determines the age-dependent birth rate into the cellular layer, and, finally, the cellular scale, which is associated to the stochastic population dynamics.
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f0005: Schematic representation of the different elements that compose our multi-scale model. We show the different levels of biological organisation as well as associated characteristic time scales (Guerrero and Alarcón, 2015) associated to each of these layers: resource scale, i.e. oxygen which is supplied at a constant rate and consumed by the cell population, cellular scale, i.e. oxygen-regulated cell cycle progression which determines the age-dependent birth rate into the cellular layer, and, finally, the cellular scale, which is associated to the stochastic population dynamics.

Mentions: The model we present in this article integrates phenomena characterised by different time scales, as schematically shown in Fig. 1. This model intends to tackle the growth and competition of cellular populations under the restriction of finite amount of available resources (in this case, oxygen) supplied at a finite rate, .


Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis
Schematic representation of the different elements that compose our multi-scale model. We show the different levels of biological organisation as well as associated characteristic time scales (Guerrero and Alarcón, 2015) associated to each of these layers: resource scale, i.e. oxygen which is supplied at a constant rate and consumed by the cell population, cellular scale, i.e. oxygen-regulated cell cycle progression which determines the age-dependent birth rate into the cellular layer, and, finally, the cellular scale, which is associated to the stochastic population dynamics.
© Copyright Policy - CC BY
Related In: Results  -  Collection

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

f0005: Schematic representation of the different elements that compose our multi-scale model. We show the different levels of biological organisation as well as associated characteristic time scales (Guerrero and Alarcón, 2015) associated to each of these layers: resource scale, i.e. oxygen which is supplied at a constant rate and consumed by the cell population, cellular scale, i.e. oxygen-regulated cell cycle progression which determines the age-dependent birth rate into the cellular layer, and, finally, the cellular scale, which is associated to the stochastic population dynamics.
Mentions: The model we present in this article integrates phenomena characterised by different time scales, as schematically shown in Fig. 1. This model intends to tackle the growth and competition of cellular populations under the restriction of finite amount of available resources (in this case, oxygen) supplied at a finite rate, .

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