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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: Partial differential equations Mathematical biology in general

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: [email protected]
MICHAEL J. WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada, email: [email protected]
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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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References

Asfahl, K. L. & Schuster, M. (2017) Social interactions in bacterial cell-cell signaling. FEMS Microbiol. Rev. 41(1), 92–107.CrossRefGoogle ScholarPubMed
Chaplain, M., Ptashnyk, M. & Sturrock, M. (2015) Hopf bifurcation in a gene regulatory network model: molecular movement causes oscillations. Math. Mod. Meth. Appl. Sci. 25(6), 1179–1215.CrossRefGoogle Scholar
Danø, S., Madsen, M. F. & Sorensøn, P. G. (2007) Quantitative characterization of cell synchronization in yeast. Proc. Nat. Acad. Sci. 104(31), 12732–12736.CrossRefGoogle Scholar
De Monte, S., d’Ovidio, F., Danø, S. & Sørensen, P. G. (2007) Dynamical quorum sensing: population density encoded in cellular dynamics. Proc. Nat. Acad. Sci. 104(47), 18377–18381.CrossRefGoogle Scholar
Dockery, J. D. & Keener, J. P. (2001) A mathematical model for quorum sensing in pseudomonas aeruginosa . Bull. Math. Biol. 63(1), 95–116.CrossRefGoogle ScholarPubMed
Ermentrout, B. (2002) Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students (software, environments and tools). SIAM Publishers. Pages: XIV + 290.Google Scholar
Fujimoto, K. & Sawai, S. (2013) A design principle of group-level decision making in cell populations. PLoS Comput. Biol. 9(6), e1003110.CrossRefGoogle ScholarPubMed
Gao, M., Zheng, H., Ren, Y., Lou, R., Wu, F., Yu, X., Liu, W. & Ma, X. (2016) A crucial role for spatial distribution in bacterial quorum sensing. Sci. Rep. 6, 34695.CrossRefGoogle ScholarPubMed
Gou, J. & Ward, M. J. (2016) An asymptotic analysis of a 2-D model of dynamically active compartments coupled by bulk diffusion. J. Nonlinear Sci. 26(4), 979–1029.CrossRefGoogle Scholar
Gregor, T., Fujimoto, K., Masaki, N. & Sawai, S. (2010) The onset of collective behavior in social amoebae. Science 328(5981), 1021–1025.CrossRefGoogle ScholarPubMed
Henson, M. A., Müller, D. & Reuss, M. (2002) Cell population modelling of yeast glycolytic oscillations. Biochem. J. 368, 433–446.CrossRefGoogle ScholarPubMed
Kamino, K., Fujimoto, K. & Sawai, S. (2011) Collective oscillations in developing cells: insights from simple systems. Develop. Growth Differ. 53, 503–517.CrossRefGoogle ScholarPubMed
Kuramoto, Y. (1975) Self-entrainment of a population of coupled non-linear oscillators. In: International Symposium on Mathematical Problems in Theoretical Physics, Springer, pp. 420–422.CrossRefGoogle Scholar
Leaman, E. J., Geuther, B. Q. & Behkam, B. (2018) Quantitative investigation of the role of intra-/intercellular dynamics in bacterial quorum sensing. ACS Synth. Biol. 7(4), 1030–1042.CrossRefGoogle ScholarPubMed
Li, B. W., Cao, X. Z. & Fu, C. (2017) Quorum sensing in populations of spatially extended chaotic oscillators coupled indirectly via a heterogeneous environment. J. Nonlinear Sci. 27(6), 1667–1686.CrossRefGoogle Scholar
Li, B. W., Fu, C., Zhang, H. & Wang, X. (2012) Synchronization and quorum sensing in an ensemble of indirectly coupled chaotic oscillators. Phys. Rev. E 86(4), 046207.CrossRefGoogle Scholar
Melke, P., Sahlin, P., Levchenko, A. & Jonsson, H. (2010) A cell-based model for quorum sensing in heterogeneous bacterial colonies. PLoS Comput. Biol. 6(6), e1000819.CrossRefGoogle ScholarPubMed
Mina, P., di Benardo, M., Savery, N. J. & Tsaneva-Atanasova, K. (2012) Modeling emergence of oscillations in communicating bacteria: a structured approach from one to many cells. J. R. Soc. Interface 10, 20120612.CrossRefGoogle Scholar
Müller, J., Kuttler, C., Hense, B. A., Rothballer, M. & Hartmann, A. (2006) Cell–cell communication by quorum sensing and dimension-reduction. J. Math. Biol. 53(4), 672–702.CrossRefGoogle ScholarPubMed
Müller, J. & Uecker, H. (2013) Approximating the dynamics of communicating cells in a diffusive medium by ODEs — homogenization with localization. J. Math. Biol. 67(5), 1023–1065.CrossRefGoogle Scholar
Nanjundiah, V. (1998) Cyclic AMP oscillations in dictyostelium discoideum: models and observations. Biophys. Chem. 72(1–2), 1–8.CrossRefGoogle ScholarPubMed
Noorbakhsh, J., Schwab, D., Sgro, A., Gregor, T. & Mehta, P. (2015) Modeling oscillations and spiral waves in Dictyostelium populations. Phys. Rev. E 91, 062711.CrossRefGoogle ScholarPubMed
Nykamp, D. Q. The idea of synchrony of phase oscillators. From Math Insight: http://mathinsight.org/synchrony_phase_oscillator_idea.Google Scholar
Rauch, E. M. & Millonas, M. (2004) The role of trans-membrane signal transduction in turing-type cellular pattern formation. J. Theor. Biol. 226, 401–407.CrossRefGoogle ScholarPubMed
Schwab, D. J., Baetica, A. & Mehta, P. (2012) Dynamical quorum-sensing in oscillators coupled through an external medium. Physica D 241(21), 1782–1788.CrossRefGoogle ScholarPubMed
Sel’Kov, E. E. (1968) Self-oscillations in glycolysis 1. A simple kinetic model. Eur. J. Biochem. 4(1), 79–86.CrossRefGoogle ScholarPubMed
Shampine, L. F. & Reichelt, M. W. (1997) The Matlab ODE suite. SIAM J. Sci. Comput. 18(1), 1–22.CrossRefGoogle Scholar
Taylor, A. F., Tinsley, M. & Showalter, K. (2015) Insights into collective cell behavior from populations of coupled chemical oscillators. Phys. Chemistry Chem Phys. 17(31), 20047–20055.CrossRefGoogle ScholarPubMed
Taylor, A. F., Tinsley, M., Wang, F., Huang, Z. & Showalter, K. (2009) Dynamical quorum sensing and synchronization in large populations of chemical oscillators. Science 323(5914), 614–6017.CrossRefGoogle ScholarPubMed
Tinsley, M. R., Taylor, A. F., Huang, Z. & Showalter, K. (2009) Emergence of collective behavior in groups of excitable catalyst-loaded particles: spatiotemporal dynamical quorum sensing. Phys. Rev. Lett. 102, 158301.CrossRefGoogle ScholarPubMed
Tinsley, M. R., Taylor, A. F., Huang, Z., Wang, F. & Showalter, K. (2010) Dynamical quorum sensing and synchronization in collections of excitable and oscillatory catalytic particles. Physica D 239(11), 785–790.CrossRefGoogle Scholar
Uecke, H., Müller, J. & Hense, B. A. (2014) Individual-based model for quorum sensing with background flow. Bull. Math. Biol. 76(7), 1727–1746.CrossRefGoogle Scholar
Ward, J. P., King, J. R., Koerber, A. J., Williams, P., Croft, J. M. & Sockett, R. E. (2001) Mathematical modelling of quorum sensing in bacteria. Math. Med. Biol. 18(3), 263–292.CrossRefGoogle Scholar
Wolf, J. & Heinrich, R. (2000) Effect of cellular interaction on glycolytic oscillations in yeast: a theoretical investigation. Biochem. J. 345, 321–334.CrossRefGoogle ScholarPubMed