Abstract
Turing instability in activator–inhibitor systems provides a paradigm of non-equilibrium self-organization; it has been extensively investigated for biological and chemical processes. Turing instability should also be possible in networks, and general mathematical methods for its treatment have been formulated previously. However, only examples of regular lattices and small networks were explicitly considered. Here we study Turing patterns in large random networks, which reveal striking differences from the classical behaviour. The initial linear instability leads to spontaneous differentiation of the network nodes into activator-rich and activator-poor groups. The emerging Turing patterns become furthermore strongly reshaped at the subsequent nonlinear stage. Multiple coexisting stationary states and hysteresis effects are observed. This peculiar behaviour can be understood in the framework of a mean-field theory. Our results offer a new perspective on self-organization phenomena in systems organized as complex networks. Potential applications include ecological metapopulations, synthetic ecosystems, cellular networks of early biological morphogenesis, and networks of coupled chemical nanoreactors.
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References
Turing, A. M. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952).
Prigogine, I. & Lefever, R. Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695–1700 (1968).
Castets, V., Dulos, E., Boissonade, J. & De Kepper, P. Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953–2956 (1990).
Ouyang, Q. & Swinney, H. L. Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352, 610–612 (1991).
Murray, J. D. Mathematical Biology (Springer, 2003).
Mikhailov, A. S. Foundations of Synergetics I. Distributed Active Systems 2nd revised edn (Springer, 1994).
Barrat, A., Barthélemy, M. & Vespignani, A. Dynamical Processes on Complex Networks (Cambridge Univ. Press, 2008).
Othmer, H. G. & Scriven, L. E. Instability and dynamic pattern in cellular networks. J. Theor. Biol. 32, 507–537 (1971).
Othmer, H. G. & Scriven, L. E. Nonlinear aspects of dynamic pattern in cellular networks. J. Theor. Biol. 43, 83–112 (1974).
Horsthemke, W., Lam, K. & Moore, P. K. Network topology and Turing instability in small arrays of diffusively coupled reactors. Phys. Lett. A 328, 444–451 (2004).
Moore, P. K. & Horsthemke, W. Localized patterns in homogeneous networks of diffusively coupled reactors. Physica D 206, 121–144 (2005).
Hanski, I. Metapopulation dynamics. Nature 396, 41–49 (1998).
Urban, D. & Keitt, T. Landscape connectivity: A graph-theoretic perspective. Ecology 82, 1205–1218 (2001).
Fortuna, M. A., Gómez-Rodrǵuez, C. & Bascompte, J. Spatial network structure and amphibian persistence in stochastic environments. Proc. R. Soc. B 273, 1429–1434 (2006).
Minor, E. S. & Urban, D. L. A graph-theory framework for evaluating landscape connectivity and conservation planning. Conserv. Biol. 22, 297–307 (2008).
Holland, M. D. & Hastings, A. Strong effect of dispersal network structure on ecological dynamics. Nature 456, 792–795 (2008).
Hufnagel, L., Brockmann, D. & Geisel, T. Forecast and control of epidemics in a globalized world. Proc. Natl Acad. Sci. USA 101, 15124–15129 (2004).
Colizza, V., Barrat, A., Barthélemy, M. & Vespignani, A. The role of the airline transportation network in the prediction and predictability of global epidemics. Proc. Natl Acad. Sci. 103, 2015–2020 (2006).
Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001).
Colizza, V., Pastor-Satorras, R. & Vespignani, A. Reaction–diffusion processes and metapopulation models in heterogeneous networks. Nature Phys. 3, 276–282 (2007).
Colizza, V. & Vespignani, A. Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations. J. Theor. Biol. 251, 450–467 (2008).
Ichinomiya, T. Frequency synchronization in a random oscillator network. Phys. Rev. E 70, 026116 (2004).
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D-U. Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006).
Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. & Zhou, C. Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008).
Meinhardt, H. & Gierer, A. Pattern formation by local self-activation and lateral inhibition. BioEssays 22, 753–760 (2000).
Harris, M. P., Williamson, S., Fallon, J. F., Meinhardt, H. & Prum, R. O. Molecular evidence for an activator–inhibitor mechanism in development of embryonic feather branching. Proc. Natl Acad. Sci. USA 102, 11734–11739 (2005).
Maini, P. K., Baker, R. E. & Chuong, C. M. The Turing model comes of molecular age. Science 314, 1397–1398 (2006).
Newman, S. A. & Bhat, R. Activator–inhibitor dynamics of vertebrate limb pattern formation. Birth Defects Res. (Part C) 81, 305–319 (2007).
Miura, T. & Shiota, K. TGFβ2 acts as an ‘activator’ molecule in reaction–diffusion model and is involved in cell sorting phenomenon in mouse limb micromass culture. Dev. Dyn. 217, 241–249 (2000).
Mimura, M. & Murray, J. D. Diffusive prey–predator model which exhibits patchiness. J. Theor. Biol. 75, 249–262 (1978).
Maron, J. L. & Harrison, S. Spatial pattern formation in an insect host-parasitoid system. Science 278, 1619–1621 (1997).
Baurmann, M., Gross, T. & Feudel, U. Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations. J. Theor. Biol. 245, 220–229 (2007).
Rietkerk, M. & van de Koppel, J. Regular pattern formation in real ecosystems. Trends Ecol. Evolut. 23, 169–175 (2008).
Barabási, A-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).
Strogatz, H. S. Exploring complex networks. Nature 410, 268–276 (2001).
Albert, R. & Barabási, A-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002).
Dorogovtsev, S. N., Goltsev, A. V., Mendes, J. F. F. & Samukhin, A. N. Spectra of complex networks. Phys. Rev. E 68, 046109 (2003).
Kim, D-H. & Motter, A. E. Ensemble averageability in network spectra. Phys. Rev. Lett. 98, 248701 (2007).
Samukhin, A. N., Dorogovtsev, S. N. & Mendes, J. F. F. Laplacian spectra of, and random walks on, complex networks: Are scale-free architectures really important? Phys. Rev. E 77, 036115 (2008).
McGraw, P. N. & Menzinger, M. Laplacian spectra as a diagnostic tool for network structure and dynamics. Phys. Rev. E 77, 031102 (2008).
Nakao, H. & Mikhailov, A. S. Diffusion-induced instability and chaos in random oscillator networks. Phys. Rev. E 79, 036214 (2009).
Cohen, R. & Havlin, S. Scale-free networks are ultrasmall. Phys. Rev. Lett. 90, 058701 (2003).
Mizuguchi, T. & Sano, M. Proportion regulation of biological cells in globally coupled nonlinear systems. Phys. Rev. Lett. 75, 966–969 (1995).
Nakajima, A. & Kaneko, K. Regulative differentiation as bifurcations of interacting cell population. J. Theor. Biol. 253, 779–787 (2008).
Karlsson, A. et al. Molecular engineering: Networks of nanotubes and containers. Nature 409, 150–152 (2001).
Bignone, F. A. Structural complexity of early embryos: A study on the nematode Caenorhabditis elegans. J. Biol. Phys. 27, 257–283 (2001).
Schnabel, R. et al. Global cell sorting in the C. elegans embryo defines a new mechanism for pattern formation. Dev. Biol. 294, 418–431 (2006).
Kondo, S. & Asai, R. A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376, 765–768 (1995).
Nakamasu, A., Takahashi, G., Kanbe, A. & Kondo, S. Interactions between zebrafish pigment cells responsible for the generation of Turing patterns. Proc. Natl Acad. Sci. USA 106, 8429–8434 (2009).
Balagaddé, F. K. et al. A synthetic Escherichia coli predator–prey ecosystem. Mol. Syst. Biol. 4, 1–8 (2008).
Acknowledgements
This work was supported by the Volkswagen Foundation, Germany, and by the MEXT, Japan (Global COE Program ‘The Next Generation of Physics, Spun from Universality and Emergence’ and Kakenhi Grant No. 19760253).
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Both authors designed the study, carried out the analysis, and contributed to writing the paper. H.N. performed numerical simulations.
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Nakao, H., Mikhailov, A. Turing patterns in network-organized activator–inhibitor systems. Nature Phys 6, 544–550 (2010). https://doi.org/10.1038/nphys1651
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DOI: https://doi.org/10.1038/nphys1651
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