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 alterna- tive 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.
Original languageEnglish
JournalJournal of Theoretical Biology
Early online date14 Sep 2018
DOIs
Publication statusE-pub ahead of print - 14 Sep 2018

Fingerprint

Cell Migration
Invasion
Cell Cycle
cell movement
Cell Movement
cell cycle
Cells
Multistage Model
Cell Proliferation
Cell proliferation
Stochastic Model
cell proliferation
Erlang Distribution
Model
Cell Division
Agent-based Model
Laplace transforms
Proliferation
Traveling Wave Solutions
Exponential distribution

Keywords

  • Cell migration
  • Travelling waves
  • Cell cycle time distribution
  • Hypoexponential distribution
  • Collective behaviour

Cite this

@article{cf0a23e5ba2a43be8d7efd8ccfc26df2,
title = "The invasion speed of cell migration models with realistic cell cycle time distributions",
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 alterna- tive 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.",
keywords = "Cell migration, Travelling waves, Cell cycle time distribution, Hypoexponential distribution, Collective behaviour",
author = "Enrico Gavagnin and Christian Yates and Timothy Rogers and Richard Mort and Matthew Ford",
note = "Copyright {\circledC} 2018. Published by Elsevier Ltd.",
year = "2018",
month = "9",
day = "14",
doi = "10.1016/j.jtbi.2018.09.010",
language = "English",
journal = "Journal of Theoretical Biology",
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T1 - The invasion speed of cell migration models with realistic cell cycle time distributions

AU - Gavagnin, Enrico

AU - Yates, Christian

AU - Rogers, Timothy

AU - Mort, Richard

AU - Ford, Matthew

N1 - Copyright © 2018. Published by Elsevier Ltd.

PY - 2018/9/14

Y1 - 2018/9/14

N2 - 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 alterna- tive 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.

AB - 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 alterna- tive 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.

KW - Cell migration

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KW - Cell cycle time distribution

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