Density-dependent quiescence in glioma invasion: instability in a simple reaction–diffusion model for the migration/proliferation dichotomy

Abstract

Gliomas are very aggressive brain tumours, in which tumour cells gain the ability to penetrate the surrounding normal tissue. The invasion mechanisms of this type of tumour remain to be elucidated. Our work is motivated by the migration/proliferation dichotomy (go-or-grow) hypothesis, i.e. the antagonistic migratory and proliferating cellular behaviours in a cell population, which may play a central role in these tumours. In this paper, we formulate a simple go-or-grow model to investigate the dynamics of a population of glioma cells for which the switch from a migratory to a proliferating phenotype (and vice versa) depends on the local cell density. The model consists of two reaction–diffusion equations describing cell migration, proliferation and a phenotypic switch. We use a combination of numerical and analytical techniques to characterize the development of spatio-temporal instabilities and travelling wave solutions generated by our model. We demonstrate that the density-dependent go-or-grow mechanism can produce complex dynamics similar to those associated with tumour heterogeneity and invasion.

Keywords:

• tumour invasion
• cell migration/proliferation
• go-or-grow
• diffusion-driven instability
• travelling wave solutions
• spatio-temporal heterogeneity

AMS Subject Classification :

• 35Q92
• 35K57
• 92C15

Acknowledgements

J.L., K.P. and X.L. acknowledge support from the National Science Foundation Division of Mathematical Sciences (DMS) and from the National Institutes of Health through grant P50GM76516 for a Centre of Excellence in Systems Biology at the University of California, Irvine. A.C., H.H. and V.C. acknowledge support from The Cullen Trust for Health Care and the National Institute for Health, Integrative Cancer Biology Program: 1U54CA149196, for the Center for Systematic Modeling of Cancer Development. V.C. also acknowledges the National Science Foundation, Division of Mathematical Sciences for grant DMS-0818104. H.B. acknowledges partial support by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).

Additional information

Notes on contributors

Helen M. Byrne

Current affiliation: Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, 24–29 St Giles, Oxford, OX1 3LB, UK