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Localised signalling compartments in 2D coupled by a bulk diffusion field: Quorum sensing and synchronous oscillations in the well-mixed limit

Part of: Mathematical biology in general Partial differential equations

Published online by Cambridge University Press:  10 August 2020

SARAFA A. IYANIWURA  and
MICHAEL J. WARD Open the ORCID record for MICHAEL J. WARD [Opens in a new window]
SARAFA A. IYANIWURA
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada, email: ward@math.ubc.ca
MICHAEL J. WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada, email: ward@math.ubc.ca
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Abstract

We analyse oscillatory instabilities for a coupled partial-ordinary differential equation (PDE-ODE) system modelling the communication between localised spatially segregated dynamically active signalling compartments that are coupled through a passive extracellular bulk diffusion field in a bounded 2D domain. Each signalling compartment is assumed to secrete a chemical into the extracellular medium (bulk region), and it can also sense the concentration of this chemical in the region around its boundary. This feedback from the bulk region, resulting from the entire collection of cells, in turn modifies the intracellular dynamics within each cell. In the limit where the signalling compartments are circular discs with a small common radius ɛ ≪ 1 and where the bulk diffusivity is asymptotically large, a matched asymptotic analysis is used to reduce the dimensionless PDE–ODE system into a nonlinear ODE system with global coupling. For Sel’kov reaction kinetics, this ODE system for the intracellular dynamics and the spatial average of the bulk diffusion field are then used to investigate oscillatory instabilities in the dynamics of the cells that are triggered due to the global coupling. In particular, numerical bifurcation software on the ODEs is used to study the overall effect of coupling defective cells (cells that behave differently from the remaining cells) to a group of identical cells. Moreover, when the number of cells is large, the Kuramoto order parameter is computed to predict the degree of phase synchronisation of the intracellular dynamics. Quorum sensing behaviour, characterised by the onset of collective behaviour in the intracellular dynamics as the number of cells increases above a threshold, is also studied. Our analysis shows that the cell population density plays a dual role of triggering and then quenching synchronous oscillations in the intracellular dynamics.

Keywords

Hopf bifurcationsynchronous oscillationsquorum sensingbulk diffusiondefector cellsKuramoto order parameterglobal coupling

MSC classification

Primary: 35B25: Singular perturbations 35B32: Bifurcation 35B36: Pattern formation
Secondary: 92B25: Biological rhythms and synchronization
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 5: 30th Anniversary Special Issue , October 2021 , pp. 1001 - 1031
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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