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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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Series of plots showing the scaling analysis of the G1/S transition age. Plot (a) shows that below the critical value rcr, which corresponds to the value of the ratio , above which there is no transition to quiescence (i.e. if  the G1/S transition age is finite when c=0),  follows an algebraic decay with a universal -independent exponent provided that the oxygen, c, is rescaled by the critical oxygen concentration, . Plot (c) shows that if, by contrast,  decays exponentially with the oxygen concentration with a characteristic concentration c0 which, provided  is larger enough than  is -independent. Plot (b) shows how ccr varies as  is changed. Similarly, plot (d) shows how  varies  changes. Parameter values as given in Table B2.
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f0040: Series of plots showing the scaling analysis of the G1/S transition age. Plot (a) shows that below the critical value rcr, which corresponds to the value of the ratio , above which there is no transition to quiescence (i.e. if the G1/S transition age is finite when c=0), follows an algebraic decay with a universal -independent exponent provided that the oxygen, c, is rescaled by the critical oxygen concentration, . Plot (c) shows that if, by contrast, decays exponentially with the oxygen concentration with a characteristic concentration c0 which, provided is larger enough than is -independent. Plot (b) shows how ccr varies as is changed. Similarly, plot (d) shows how varies changes. Parameter values as given in Table B2.

Mentions: We can see this regularity in Fig. 8. Fig. 8(a) shows as a function of the ratio , for six different values of , where is the critical value. We see that all graphs fall together on a straight line in log–log space, indicating a power-law dependence of on . On the other hand, Fig. 8(c) shows the dependence of the normalized function , on c, for five values . Here, . We see that there is good agreement between the five values for small c, with the disagreement increasing with increasing c. Thus, a good scaling approximation for is given by(13)aG1/S(c,p6p3)≃{a+(p6p3)e−c/c0ifp6p3>rcra−(cccr(p6/p3)−1)−βifp6p3<rcrHere, c0, , and β are constants. According to our analysis of the data presented in Fig. 8, these constants are given by , and . Likewise, is obtained by fitting to the data presented in Fig. 8(d). The critical oxygen concentration for quiescence ccr can be estimated analytically (with parameter values taken from Table B2):(14)ccr(p6p3)=1−1β1log(1a3H0(a1+a2d2d1[e2f]t(1−11−a0(p3p6)2)))where a1, a3, d1, d2, , and H0 are parameters defined in Appendix B, Table B2. The parameter a0 can be estimated as follows. Let A(c) be defined as:(15)A(c)=(p6p3)2(1−d2(d2+d1a1−a3[H]a2)[e2f]t)where . , where cbif is the critical value of the oxygen concentration for which the saddle node bifurcation occurs, i.e. the critical value for which the number of real, positive solutions of the equation:(16)e1(1−x)(J4+x)(b2+b3x)=A(c)e2b1m*[e2f]tx(1+J3−x)goes from 3 to 1. The parameters bi, ei, and Ji are defined in Appendix B, Table B2


Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis
Series of plots showing the scaling analysis of the G1/S transition age. Plot (a) shows that below the critical value rcr, which corresponds to the value of the ratio , above which there is no transition to quiescence (i.e. if  the G1/S transition age is finite when c=0),  follows an algebraic decay with a universal -independent exponent provided that the oxygen, c, is rescaled by the critical oxygen concentration, . Plot (c) shows that if, by contrast,  decays exponentially with the oxygen concentration with a characteristic concentration c0 which, provided  is larger enough than  is -independent. Plot (b) shows how ccr varies as  is changed. Similarly, plot (d) shows how  varies  changes. Parameter values as given in Table B2.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC5016039&req=5

f0040: Series of plots showing the scaling analysis of the G1/S transition age. Plot (a) shows that below the critical value rcr, which corresponds to the value of the ratio , above which there is no transition to quiescence (i.e. if the G1/S transition age is finite when c=0), follows an algebraic decay with a universal -independent exponent provided that the oxygen, c, is rescaled by the critical oxygen concentration, . Plot (c) shows that if, by contrast, decays exponentially with the oxygen concentration with a characteristic concentration c0 which, provided is larger enough than is -independent. Plot (b) shows how ccr varies as is changed. Similarly, plot (d) shows how varies changes. Parameter values as given in Table B2.
Mentions: We can see this regularity in Fig. 8. Fig. 8(a) shows as a function of the ratio , for six different values of , where is the critical value. We see that all graphs fall together on a straight line in log–log space, indicating a power-law dependence of on . On the other hand, Fig. 8(c) shows the dependence of the normalized function , on c, for five values . Here, . We see that there is good agreement between the five values for small c, with the disagreement increasing with increasing c. Thus, a good scaling approximation for is given by(13)aG1/S(c,p6p3)≃{a+(p6p3)e−c/c0ifp6p3>rcra−(cccr(p6/p3)−1)−βifp6p3<rcrHere, c0, , and β are constants. According to our analysis of the data presented in Fig. 8, these constants are given by , and . Likewise, is obtained by fitting to the data presented in Fig. 8(d). The critical oxygen concentration for quiescence ccr can be estimated analytically (with parameter values taken from Table B2):(14)ccr(p6p3)=1−1β1log(1a3H0(a1+a2d2d1[e2f]t(1−11−a0(p3p6)2)))where a1, a3, d1, d2, , and H0 are parameters defined in Appendix B, Table B2. The parameter a0 can be estimated as follows. Let A(c) be defined as:(15)A(c)=(p6p3)2(1−d2(d2+d1a1−a3[H]a2)[e2f]t)where . , where cbif is the critical value of the oxygen concentration for which the saddle node bifurcation occurs, i.e. the critical value for which the number of real, positive solutions of the equation:(16)e1(1−x)(J4+x)(b2+b3x)=A(c)e2b1m*[e2f]tx(1+J3−x)goes from 3 to 1. The parameters bi, ei, and Ji are defined in Appendix B, Table B2

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