No abstract

We analyse the effect of intrinsic fluctuations on the properties of bistable stochastic systems with time scale separation operating under quasi-steady state conditions. We first formulate a stochastic generalisation of the quasi-steady state approximation based on the semi-classical approximation of the partial differential equation for the generating function associated with the chemical master equation. Such approximation proceeds by optimising an action functional whose associated set of Euler-Lagrange (Hamilton) equations provides the most likely fluctuation path. We show that, under appropriate conditions granting time scale separation, the Hamiltonian can be re-scaled so that the set of Hamilton equations splits up into slow and fast variables, whereby the quasi-steady state approximation can be applied. We analyse two particular examples of systems whose mean-field limit has been shown to exhibit bi-stability: an enzyme-catalysed system of two mutually inhibitory proteins and a gene regulatory circuit with self-activation. Our theory establishes that the number of molecules of the conserved species is order parameters whose variation regulates bistable behaviour in the associated systems beyond the predictions of the mean-field theory. This prediction is fully confirmed by direct numerical simulations using the stochastic simulation algorithm. This result allows us to propose strategies whereby, by varying the number of molecules of the three conserved chemical species, cell properties associated to bistable behaviour (phenotype, cell-cycle status, etc.) can be controlled.

No 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.