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Calcium Waves in Thin Visco-Elastic Cells

Published online by Cambridge University Press:  12 June 2013

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Abstract

The model we consider treats the cell as a viscoelastic medium filling one of two kinds of thin domains (“shapes” of cells): the thin slab being a caricature of a tissue and the thin circular cylinder mimicking a long cell. This enables us to simplify the system of mechano-chemical equations. We construct abundant classes of explicit, but approximate, formulae for heteroclinic solutions to these equations.

Type
Research Article
Copyright
© EDP Sciences, 2013

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References

Murray, J. D., Oster, G. F.. Generation of biological pattern and form. IMA Journal of Mathematics Applied in Medicine and Biology, 1 (1984), 51-75. CrossRefGoogle ScholarPubMed
D, Lane, C., Murray, J. D., Manoranjan, V. S.. Analysis of wave phenomena in a morphogenic mechanochemical model and an application to post-fertilization waves in eggs. IMA Journal of Mathematics Applied in Medicine and Biology, 4 (1987), 309-331. CrossRefGoogle Scholar
Brière, C., Goodwin, B.C.. Effects of calcium input/output on the stability of a system for calcium-regulated viscoelastic strain fields, Journal of Mathematical Biology, vol. 28, 585-593, 1990.Hart CrossRefGoogle ScholarPubMed
Kaźmierczak, B., Dyzma, M.. Mechanical effects coupled with calcium waves. Archives of Mechanics, 62(2) (2010), 121-133. Google Scholar
Kaźmierczak, B., Peradzyński, Z.. Calcium waves with fast buffers and mechanical effects, Journal of Mathematical Biology, 62 (2011), 1-38. CrossRefGoogle Scholar
Flores, G., Minzoni, A., Mischaikov, K., Moll, V.. Post-fertilization travelling waves on eggs, Nonlinear Analysis: Theory, Methods and Applications. 36(1) (1999), 45-62. Google Scholar
J. D. Murray. Mathematical Biology. 2nd ed. Springer, Berlin, 1993.
J. Keener, J. Sneyd. Mathematical Physiology. Springer, New York, 1998
P. Fife. Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics, Vo. 28, Springer, New York, 1979.
Kaźmierczak, B., Volpert, V.. Existence of heteroclinic orbits for systems satisfying monotonicity conditions. Nonlinear Analysis, 55 (2003), 467-491 CrossRefGoogle Scholar
Kaźmierczak, B., Volpert, V.. Travelling calcium waves in systems with non diffusing buffers. Math. Mod. Meth. Appl. Sci., 18 (2008), 883-912. CrossRefGoogle Scholar
Kaźmierczak, B., Volpert, V.. Calcium waves in systems with immobile buffers a a limit of waves for systems with non zero diffusion. Nonlinearity 21 (2008), 71-96 CrossRefGoogle Scholar
Richards, F. J.. A flexible growth function for empirical use. Journal of Experimental Botany, 10(1959), 290-300. CrossRefGoogle Scholar
Peradzyński, Z.. Diffusion of calcium in biological tissues and accompanying mechano-chemical effects, Archives of Mechanics, 62(2) (2010), 423-440. Google Scholar
Piechór, K.. Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment. Archives of Mechanics, 64(5) (2012), 477-509. Google Scholar