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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.


This figure shows stochastic simulation results regarding the variation of the probability of fixation of the quiescence population varies the efficiency of a cell-cycle dependent therapy changes. The efficiency of the therapy is measured in terms of the survival fraction FS. The orange vertical line represents the mean-field, theoretical critical survival fraction (Eq. 54). We observe that as the carrying capacity of the system increases, which in these simulations is achieved by increasing the rate of oxygen supply, S, the results of the stochastic simulations tend towards the mean-field prediction. 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. Each data point corresponds to the average over 500 realisations of the stochastic population dynamics. (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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f0065: This figure shows stochastic simulation results regarding the variation of the probability of fixation of the quiescence population varies the efficiency of a cell-cycle dependent therapy changes. The efficiency of the therapy is measured in terms of the survival fraction FS. The orange vertical line represents the mean-field, theoretical critical survival fraction (Eq. 54). We observe that as the carrying capacity of the system increases, which in these simulations is achieved by increasing the rate of oxygen supply, S, the results of the stochastic simulations tend towards the mean-field prediction. 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. Each data point corresponds to the average over 500 realisations of the stochastic population dynamics. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Mentions: Fig. 13 shows simulation results for the variation of probability of fixation of the quiescent population as the survival fraction of the therapy, FS, changes. Our simulation results show qualitative agreement with our mean-field theory (see 6.1, 6.2): as the survival fraction increases (i.e. the therapy becomes less efficient), the probability of fixation abruptly decreases from almost certainty of fixation to almost certainty of extinction. We observe that our mean-field theoretical predicts a critical value for FS slightly smaller than the observed when fluctuations due to finite size effects are present. However, we observe that, as the carrying capacity of the system is increased, the critical value of FS converges to the mean-field value.


Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis
This figure shows stochastic simulation results regarding the variation of the probability of fixation of the quiescence population varies the efficiency of a cell-cycle dependent therapy changes. The efficiency of the therapy is measured in terms of the survival fraction FS. The orange vertical line represents the mean-field, theoretical critical survival fraction (Eq. 54). We observe that as the carrying capacity of the system increases, which in these simulations is achieved by increasing the rate of oxygen supply, S, the results of the stochastic simulations tend towards the mean-field prediction. 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. Each data point corresponds to the average over 500 realisations of the stochastic population dynamics. (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

f0065: This figure shows stochastic simulation results regarding the variation of the probability of fixation of the quiescence population varies the efficiency of a cell-cycle dependent therapy changes. The efficiency of the therapy is measured in terms of the survival fraction FS. The orange vertical line represents the mean-field, theoretical critical survival fraction (Eq. 54). We observe that as the carrying capacity of the system increases, which in these simulations is achieved by increasing the rate of oxygen supply, S, the results of the stochastic simulations tend towards the mean-field prediction. 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. Each data point corresponds to the average over 500 realisations of the stochastic population dynamics. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Mentions: Fig. 13 shows simulation results for the variation of probability of fixation of the quiescent population as the survival fraction of the therapy, FS, changes. Our simulation results show qualitative agreement with our mean-field theory (see 6.1, 6.2): as the survival fraction increases (i.e. the therapy becomes less efficient), the probability of fixation abruptly decreases from almost certainty of fixation to almost certainty of extinction. We observe that our mean-field theoretical predicts a critical value for FS slightly smaller than the observed when fluctuations due to finite size effects are present. However, we observe that, as the carrying capacity of the system is increased, the critical value of FS converges to the mean-field value.

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.