Stochastic multiscale models of cell populationdynamics: Asymptotic and numerical methods

In this paper we present a new methodology that allows us to formulate and analysestochastic multiscale models of the dynamics of cell populations. In the spirit of existing hybridmultiscale models, we set up our model in a hierarchical way according to the characteristictime scales involved, where the stochastic population dynamics is governed by the birth anddeath rates as prescribed by the corresponding intracellular pathways (e.g. stochastic cell-cyclemodel). The feed-back loop is closed by the coupling betweenthe dynamics of the populationand the intracellular dynamics via the concentration of oxygen: Cells consume oxygen which,in turn, regulate the rate at which cells proceed through their cell-cycle. The coupling betweenintracellular and population dynamics is carried out through a novel method to obtain the birthrate from the stochastic cell-cycle model, based on a mean-first passage time approach. Cellproliferation is assumed to be activated when one or more of the proteins involved in the cell-cycle regulatory pathway hit a threshold. This view allows us to calculate the birth rate as afunction of the age of the cell and the extracellular oxygen in terms of the corresponding mean-first passage time. We then proceed to formulate the stochastic dynamics of the population ofcells in terms of an age-structured Master Equation. Further, we have developed generalisationsof asymptotic (WKB) methods for our age-structured Master Equation as well as aτ−leapmethod to simulate the evolution of our age-structured population. Finally, we illustrate thisgeneral methodology with a particular example of a cell population where progression throughthe cell-cycle is regulated by the availability of oxygen