Stochastic Multiscale Models of Cell Population Dynamics: Asymptotic and Numerical Methods

¹ Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
² Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Spain
³ Departament de Matemàtiques, Universitat Atonòma de Barcelona, 08193 Bellaterra, Barcelona, Spain

^⋆ Corresponding author. E-mail: pguerrero@ucl.ac.uk

Abstract

In this paper we present a new methodology that allows us to formulate and analyse stochastic multiscale models of the dynamics of cell populations. In the spirit of existing hybrid multiscale models, we set up our model in a hierarchical way according to the characteristic time scales involved, where the stochastic population dynamics is governed by the birth and death rates as prescribed by the corresponding intracellular pathways (e.g. stochastic cell-cycle model). The feed-back loop is closed by the coupling between the dynamics of the population and 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 between intracellular and population dynamics is carried out through a novel method to obtain the birth rate from the stochastic cell-cycle model, based on a mean-first passage time approach. Cell proliferation 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 a function 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 of cells in terms of an age-structured Master Equation. Further, we have developed generalisations of asymptotic (WKB) methods for our age-structured Master Equation as well as a τ −leap method to simulate the evolution of our age-structured population. Finally, we illustrate this general methodology with a particular example of a cell population where progression through the cell-cycle is regulated by the availability of oxygen.