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Effective nonlocal kernels on reaction-diffusion networks. (English) Zbl 1457.92066

Summary: A new method to derive an essential integral kernel from any given reaction-diffusion network is proposed. Any network describing metabolites or signals with arbitrary many factors can be reduced to a single or a simpler system of integro-differential equations called “effective equation” including the reduced integral kernel (called “effective kernel”) in the convolution type. As one typical example, the Mexican hat shaped kernel is theoretically derived from two component activator-inhibitor systems. It is also shown that a three component system with quite different appearance from activator-inhibitor systems is reduced to an effective equation with the Mexican hat shaped kernel. It means that the two different systems have essentially the same effective equations and that they exhibit essentially the same spatial and temporal patterns. Thus, we can identify two different systems with the understanding in unified concept through the reduced effective kernels. Other two applications of this method are also given: Applications to pigment patterns on skins (two factors network with long range interaction) and waves of differentiation (called proneural waves) in visual systems on brains (four factors network with long range interaction). In the applications, we observe the reproduction of the same spatial and temporal patterns as those appearing in pre-existing models through the numerical simulations of the effective equations.

MSC:

92C42 Systems biology, networks
92C40 Biochemistry, molecular biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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[1] Amari, S., Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern., 27, 77-87 (1977) · Zbl 0367.92005
[2] Bates, P. W.; Chen, F., Spectral analysis of traveling waves for nonlocal evolution equations, SIAM J. Math. Anal., 38, 1, 116-126 (2006) · Zbl 1134.35060
[3] Bates, P. W.; Chen, X.; Chmaj, A. J.J., Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions, Calc. Var., 24, 3, 261-281 (2005) · Zbl 1086.49008
[4] Carrillo, J. A.; Murakawa, H.; Sato, M.; Togashi, H.; Trush, O., A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation, J. Theor. Biol., 474, 14-24 (2019) · Zbl 1414.92133
[5] Corless, R. M.; Gonnet, G. H.; Hare, D. E.G.; Jeffrey, D. J.; Knuth, D. E., On the Lambert W function, Adv. Comput. Math., 5, 329-359 (1996) · Zbl 0863.65008
[6] Ei, S.-I., Sato, M., Tanaka, Y., Yasugi, T. Continuous method for spatial discrete models with nonlocal interactions remaining cell or lattice size (submitted).
[7] Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Kybernetik, 12, 30-39 (1972) · Zbl 1434.92013
[8] Kondo, S., An updated kernel-based Turing model for studying the mechanisms of biological pattern formation, J. Theor. Biol., 414, 120-127 (2017)
[9] Kondo, S.; Asai, R., A reaction-diffusion wave on the skin of the marine angelfsh Pomacanthus, Nature, 376, 765-768 (1995)
[10] Kuffler, S. W., Discharge patterns and functional organization of mammalian retina, J. Neurophysiol., 16, 37-68 (1953)
[11] Marcon, L., Diego, X., Sharpe, J., Muller, P., 2016. High-throughput mathematical analysis identifies Turing networks for patterning with equally diffusing signals. eLife 5. doi: 10.7554/eLife.14022.
[12] Meinhardt, H., Models of Biological Pattern Formation (1982), Academic Press: Academic Press London
[13] Miura, T., Modulation of activator diffusion by extracellular matrix in Turing system, RIMS Kyokaku Bessatsu B3, 12 (2007) · Zbl 1417.92026
[14] Murray, J., Mathematical Biology (2001), Springer: Springer USA
[15] Nakamasu, A., Takahashi, G., Kanbe, A., Kondo, S., 2009. Interactions between zebrafish pigment cells responsible for the generation of Turing patterns. PNAS 106 (21), 8429-8434.
[16] Oster, G. F., Lateral inhibition models of developmental processes, Math. Biosci., 90, 256-286 (1988) · Zbl 0651.92001
[17] Ouyang, Q.; Swinney, H., Transition from a uniform state to hexagonal and striped Turing patterns, Nature, 352, 3 (1991)
[18] Painter, K. J.; Bloomfield, J. M.; Sherratt, J. A.; Gerisch, A., A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bull. Math. Biol., 77, 6, 1132-1165 (2015) · Zbl 1335.92026
[19] Sato, M.; Yasugi, T.; Minami, Y.; Miura, T.; Nagayama, M., Notch-mediated lateral inhibition regulates proneural wave propagation when combined with EGF-mediated reaction diffusion, Proc. Natl. Acad. Sci., 113, 35, E5153-E5162 (2016)
[20] Sushida, T., Kondo, S., Sugihara, K., Mimura, M., 2018. A differential equation model of retinal processing for understanding lightness optical illusions. Jpn. J. Ind. Appl. Math. 35 (1), 117-156. · Zbl 1390.93134
[21] Turing, A. M., The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. Ser. B, 237, 37-72 (1952) · Zbl 1403.92034
[22] Watanabe, M.; Kondo, S., Is pigment patterning in fish skin determined by the Turing mechanism, Trends Genet., 31, 2, 88-96 (2015)
[23] Yamaguchi, M.; Yoshimoto, E.; Kondo, S., Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism, Proc. Natl. Acad. Sci. USA, 104, 4790-4793 (2007)
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