zbMATH — the first resource for mathematics

An asymptotic analysis of a 2-D model of dynamically active compartments coupled by bulk diffusion. (English) Zbl 1439.92024
Summary: A class of coupled cell-bulk ODE-PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius \(\varepsilon\ll 1\) are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell-bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE-PDE cell-bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell-bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator-inhibitor reaction-diffusion systems in the limit of long-range inhibition and short-range activation.

92B25 Biological rhythms and synchronization
35A08 Fundamental solutions to PDEs
35B20 Perturbations in context of PDEs
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
35M33 Initial-boundary value problems for mixed-type systems of PDEs
Full Text: DOI arXiv
[1] Busenberg, SN; Mahaffy, JM, A compartmental reaction-diffusion cell cycle model, Comput. Math. Appl., 18, 883-892, (1989) · Zbl 0722.92008
[2] Busenberg, SN; Mahaffy, JM, The effects of dimension and size for a compartmental model of repression, SIAM J. Appl. Math., 48, 882-903, (1988) · Zbl 0662.92011
[3] Chen, W; Ward, MJ, The stability and dynamics of localized spot patterns in the two-dimensional gray-Scott model, SIAM J. Appl. Dyn. Syst., 10, 582-666, (2011) · Zbl 1223.35033
[4] Chiang, WY; Li, YX; Lai, PY, Simple models for quorum sensing: nonlinear dynamical analysis, Phys. Rev. E., 84, 041921, (2011)
[5] Monte, S; d’Ovido, F; Dano, S; Sørensen, PG, Dynamical quorum sensing: population density encoded in cellular dynamics, Proc. Natl. Acad. Sci., 104, 18377-18381, (2007)
[6] Ermentrout, G.B.: Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. SIAM 2002, Philadelphia, USA · Zbl 1003.68738
[7] Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge University Press, Cambridge (1990) · Zbl 0837.92009
[8] Gomez-Marin, A; Garcia-Ojalvo, J; Sancho, JM, Self-sustained spatiotemporal oscillations induced by membrane-bulk coupling, Phys. Rev. Lett., 98, 168303, (2007)
[9] Gou, J., Li, Y.X., Nagata, W.: Interactions of in-phase and anti-phase synchronies in two cells coupled by a spatially diffusing chemical: double-hopf bifurcations, submitted. IMA J. Appl. Math. p. 23 (2015) · Zbl 1401.92014
[10] Gou, J., Ward, M.J.: Oscillatory dynamics for a coupled membrane-bulk diffusion model with Fitzhugh-Nagumo kinetics. SIAM J. Appl. Math. p. 23 (2015) · Zbl 1347.35032
[11] Gou, J; Li, YX; Nagata, W; Ward, MJ, Synchronized oscillatory dynamics for a 1-D model of membrane kinetics coupled by linear bulk diffusion, SIAM J. Appl. Dyn. Syst., 14, 2096-2137, (2015) · Zbl 1331.35038
[12] Gou, J., Chiang, W.Y., Lai, P.Y., Ward, M.J., Li, Y.X.: A theory of synchrony by coupling through a diffusive chemical signal. Submitted. Phys. D p. 28 (2016) · Zbl 1376.92020
[13] Gregor, T; Fujimoto, K; Masaki, N; Sawai, S, The onset of collective behavior in social amoeba, Science, 328, 1021-1025, (2010)
[14] Krsmanovic, LZ; Mores, N; Navarro, CE; Arora, KK; Catt, KJ, An agonist-induced switch in g protein coupling of the gonadotropin-releasing hormone receptor regulates pulsatile neuropeptide secretion, Proc. Natl. Acad. Sci. USA, 100, 2969-2974, (2003)
[15] Kropinski, MC; Quaife, BD, Fast integral equation methods for the modified Helmholtz equation, J. Comput. Phys., 230, 425-434, (2011) · Zbl 1207.65144
[16] Kurella, V; Tzou, J; Coombs, D; Ward, MJ, Asymptotic analysis of first passage time problems inspired by ecology, Bull. Math Biol., 77, 83-125, (2015) · Zbl 1319.35275
[17] Levy, C; Iron, D, Dynamics and stability of a three-dimensional model of cell signal transduction, J. Math. Biol., 67, 1691-1728, (2014) · Zbl 1283.35151
[18] Levy, C; Iron, D, Dynamics and stability of a three-dimensional model of cell signal transduction with delay, Nonlinearity, 28, 2515-2553, (2015) · Zbl 1356.35265
[19] Li, YX; Khadra, A, Robust synchrony and rhythmogenesis in endocrine neurons via autocrine regulations in vitro and in vivo, Bull. Math. Biol., 70, 2103-2125, (2008) · Zbl 1170.92006
[20] Müller, J; Kuttler, C; Hense, BA; Rothballer, M; Hartmann, A, Cell-cell communication by quorum sensing and dimension-reduction, J. Math. Biol., 53, 672-702, (2006) · Zbl 1113.92022
[21] Müller, J; Uecker, H, Approximating the dynamics of communicating cells in a diffusive medium by ODEs: homogenization with localization, J. Math. Biol., 67, 1023-1065, (2013) · Zbl 1277.35036
[22] Naqib, F; Quail, T; Musa, L; Vulpe, H; Nadeau, J; Lei, J; Glass, L, Tunable oscillations and chaotic dynamics in systems with localized synthesis, Phys. Rev. E, 85, 046210, (2012)
[23] Nanjundiah, V, Cyclic AMP oscillations in dictyostelium discoideum: models and observations, Biophys. Chem., 72, 1-8, (1998)
[24] Noorbakhsh, J; Schwab, D; Sgro, A; Gregor, T; Mehta, P, Modeling oscillations and spiral waves in dictyostelium populations, Phys. Rev. E, 91, 062711, (2015)
[25] Novak, B; Tyson, JJ, Design principles of biochemical oscillators, Nat. Rev. Mol. Cell Biol., 9, 981-991, (2008)
[26] Peirce, AP; Rabitz, H, Effect of defect structures on chemically active surfaces: a continuum approach, Phys. Rev. B., 38, 1734-1753, (1998)
[27] Pillay, S; Ward, MJ; Pierce, A; Kolokolnikov, T, An asymptotic analysis of the Mean first passage time for narrow escape problems: part I: two-dimensional domains, SIAM Multiscale Model. Simul., 8, 803-835, (2010) · Zbl 1203.35023
[28] Rauch, EM; Millonas, M, The role of trans-membrane signal transduction in Turing-type cellular pattern formation, J. Theor. Biol., 226, 401-407, (2004)
[29] Riecke, H; Kramer, L, Surface-induced chemical oscillations and their influence on space- and time-periodic patterns, J. Chem. Phys., 83, 3941, (1985)
[30] Rozada, I; Ruuth, S; Ward, MJ, The stability of localized spot patterns for the Brusselator on the sphere, SIAM J. Appl. Dyn. Syst., 13, 564-627, (2014) · Zbl 1302.35033
[31] Schwab, DJ; Baetica, A; Mehta, P, Dynamical quorum-sensing in oscillators coupled through an external medium, Phys. D, 241, 1782-1788, (2012) · Zbl 1401.92014
[32] Taylor, AF; Tinsley, M; Wang, F; Huang, Z; Showalter, K, Dynamical quorum sensing and synchronization in large populations of chemical oscillators, Science, 323, 6014-617, (2009)
[33] Taylor, AF; Tinsley, M; Showalter, K, Insights into collective cell behavior from populations of coupled chemical oscillators, Phys. Chem. Chem. Phys., 17, 20047-20055, (2015)
[34] Tinsley, MR; Taylor, AF; Huang, Z; Wang, F; Showalter, K, Dynamical quorum sensing and synchronization in collections of excitable and oscillatory catalytic particles, Phys. D, 239, 785-790, (2010)
[35] Tinsley, MR; Taylor, AF; Huang, Z; Showalter, K, Emergence of collective behavior in groups of excitable catalyst-loaded particles: spatiotemporal dynamical quorum sensing, Phys. Rev. Lett., 102, 158301, (2009)
[36] Ward, M.J.: Asymptotics for strong localized perturbations: theory and applications. Online lecture notes for fourth winter school on applied mathematics, CityU of Hong Kong, p. 100 (2010) · Zbl 1141.35345
[37] Wei, J; Winter, M, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Sci., 11, 415-458, (2001) · Zbl 1141.35345
[38] Wei, J; Winter, M, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57, 53-89, (2008) · Zbl 1141.92007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.