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.
Highlights
In reference [1], it was assumed that the response of this transition to hypoxia is mediated by the protein p27, an element of the cyclindependent protein kinases (Cdks) network whose production is upregulated under hypoxia [16,18], recent studies cast some doubts on the role of p27 as the mediator of hypoxic effects on cell-cycle progression [9, 23]
We present a formulation of a stochastic multiscale model of cell population dynamics where different levels of biological organisation, characterised by different time scales are coupled
We describe the stochastic cellular population dynamics by means of a birth and death process where the birth rate is determined by a model of oxygen-regulated cell cycle progression, coupling the intracellular and cellular scales
Summary
Multiscale modelling of biological systems has become a very active field of research with significant contributions being made in a number of different areas from cardiology [25, 31, 36, 52] to developmental biology [26, 40, 49, 51, 62] and tumour growth [2, 11, 14, 15, 27, 33, 34, 41,42,43,44,45, 47, 48, 55,56,57]. In order to try and address these issues, a great deal of research has been done in the development and analysis of multiscale models Such models are capable of incorporating within a single model different sub-models corresponding to different levels of biological organisation (intracellular, cell-to-cell interaction, whole-tissue level, etc.), which are usually characterised by diverse time and length scales, and the coupling between them, so that the global tissue behaviour can be analysed as an emergent property of the different coupled elements [11, 14, 33, 41, 47, 56]. Hybrid models have been proposed to study different aspects of tumour growth such as response to therapy [2, 48], tumour-induced angiogenesis [34, 42, 43] and evolutionary dynamics of tumour growth [47] Another possible approach to multiscale modelling is to use multi-phase models [10, 33, 45, 57].
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