An equivalence framework for an age-structured multi-stage representation of the cell cycle

TL;DR

This paper introduces a theoretical framework linking stochastic and deterministic models of age-structured cell populations, providing new tools for understanding cell cycle dynamics and spatial pattern formation.

Contribution

It develops a hierarchical system of equations for an age-structured multi-stage Markov process and shows its mean behaviour aligns with classical PDE models, extending to spatial contexts.

Findings

Equivalence between stochastic and deterministic models established.

Framework applicable to spatial and reaction-diffusion processes.

Facilitates modeling of cell cycle and pattern formation.

Abstract

We develop theoretical equivalences between stochastic and deterministic models for populations of individual cells stratified by age. Specifically, we develop a hierarchical system of equations describing the full dynamics of an age-structured multi-stage Markov process for approximating cell cycle time distributions. We further demonstrate that the resulting mean behaviour is equivalent, over large timescales, to the classical McKendrick-von Foerster integro-partial differential equation. We conclude by extending this framework to a spatial context, facilitating the modelling of travelling wave phenomena and cell-mediated pattern formation. More generally, this methodology may be extended to myriad reaction-diffusion processes for which the age of individuals is relevant to the dynamics.