The invasion speed of cell migration models with realistic cell cycle time distributions

TL;DR

This paper links realistic cell cycle time distributions, modeled via multi-stage processes, to the invasion speed of cell migration, providing analytical tools to predict how cell cycle variability influences invasion dynamics.

Contribution

It introduces a mathematical framework connecting multi-stage cell cycle models to invasion speed, including an analytical expression for N-stage Erlang-distributed cell cycle times.

Findings

Invasion speed depends on the Laplace transform of the CCTD.

Erlang distribution yields the minimum invasion speed.

Exponential distribution yields the maximum invasion speed.

Abstract

Cell proliferation is typically incorporated into stochastic mathematical models of cell migration by assuming that cell divisions occur after an exponentially distributed waiting time. Experimental observations, however, show that this assumption is often far from the real cell cycle time distribution (CCTD). Recent studies have suggested an alternative approach to modelling cell proliferation based on a multi-stage representation of the CCTD. In order to validate and parametrise these models, it is important to connect them to experimentally measurable quantities. In this paper we investigate the connection between the CCTD and the speed of the collective invasion. We first state a result for a general CCTD, which allows the computation of the invasion speed using the Laplace transform of the CCTD. We use this to deduce the range of speeds for the general case. We then focus on the more realistic case of multi-stage models, using both a stochastic agent-based model and a set of reaction-diffusion equations for the cells' average density. By studying the corresponding travelling wave solutions, we obtain an analytical expression for the speed of invasion for a general N-stage model with identical transition rates, in which case the resulting cell cycle times are Erlang distributed. We show that, for a general N-stage model, the Erlang distribution and the exponential distribution lead to the minimum and maximum invasion speed, respectively. This result allows us to determine the range of possible invasion speeds in terms of the average proliferation time for any multi-stage model.