# American Institute of Mathematical Sciences

2013, 10(3): 743-759. doi: 10.3934/mbe.2013.10.743

## Calcium waves with mechano-chemical couplings

 1 Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland 2 Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Received  June 2012 Revised  November 2012 Published  April 2013

As follows from experiments, waves of calcium concentration in biological tissues can be easily excited by a local mechanical stimulation. Therefore the complete theory of calcium waves should also take into account coupling between mechanical and chemical processes. In this paper we consider the existence of travelling waves for buffered systems, as in [22], completed, however, by an equation for mechanical equilibrium and respective mechanochemical coupling terms. Thus the considered, coupled system consists of reaction-diffusion equations (for the calcium and buffers concentrations) and equations for the balance of mechanical forces.
Citation: Bogdan Kazmierczak, Zbigniew Peradzynski. Calcium waves with mechano-chemical couplings. Mathematical Biosciences & Engineering, 2013, 10 (3) : 743-759. doi: 10.3934/mbe.2013.10.743
##### References:
 [1] D. Bia, R. Armentano, D. Craiem, J. Grignola, F. Gines, A. Simon and J. Levenson, Smooth muscle role on pulmonary arterial function during acute pulmonary hypertension in sheep,, Acta Physiol. Scand., 181 (2004), 359. doi: 10.1111/j.1365-201X.2004.01294.x. [2] E. C. M. Crooks, On the Vol'pert theory of travelling wave solutions for parabolic equations,, Nonlinear Analysis TM & A, 26 (1996), 1621. doi: 10.1016/0362-546X(95)00038-W. [3] P. B. Dobrin and J. M. Doyle, Vascular smooth muscle and the anisotropy of dog carotid artery,, Circ. Res., 27 (1970), 105. doi: 10.1161/01.RES.27.1.105. [4] A. Doyle, W. Marganski and J. Lee, Calcium transients induce spatially coordinated increases in traction force during the movement of fish keratocytes,, Journal of Cell Science, 117 (2004), 2203. doi: 10.1242/jcs.01087. [5] M. Falcke, Reading the patterns in living cells - the physics of $Ca^{2+}$ signaling,, Advances in Physics, 53 (2004), 255. [6] G. Flores, A. Minzoni, K. Mischaikov and V. Moll, Post-fertilization travelling waves on eggs,, Nonlinear Analysis: Theory, 36 (1999), 45. doi: 10.1016/S0362-546X(97)00696-2. [7] D. Hong, D. Jaron, D. G. Buerk and K. A. Barbee, Heterogeneous response of microvascular endothelial cells to shear stress,, Am. J. Physiol. (Heart Circ. Physiol.), 290 (2006), 2498. doi: 10.1152/ajpheart.00828.2005. [8] L. F. Jaffe, The path of calcium in cytosolic calcium oscillations: A unifying hypothesis,, Proc. Natl. Acad. Sci. USA, 88 (1991), 9883. doi: 10.1073/pnas.88.21.9883. [9] L. F. Jaffe, Stretch-activated calcium channels relay fast calcium waves propagated by calcium-induced calcium influx,, Biol. Cell, 99 (2007), 75. doi: 10.1042/BC20060031. [10] N. R. Jorgensen, S. C. Teilmann, Z. Henriksen, R. Civitelli, O. H. Sorensen and T. H. Steinberg, Activation of L-type calcium channels is required for gap junction-mediated intercellular calcium signaling in osteoblastic cells,, J. Biological Chemistry, 278 (2003), 4082. doi: 10.1074/jbc.M205880200. [11] B. Kazmierczak and Z. Peradzynski, Calcium waves with fast buffers and mechanical effects,, Journal of Mathematical Biology, 62 (2011), 1. doi: 10.1007/s00285-009-0323-2. [12] B. Kazmierczak and V. Volpert, Calcium waves in systems with immobile buffers as a limit of waves for systems with non zero diffusion,, Nonlinearity, 21 (2008), 71. doi: 10.1088/0951-7715/21/1/004. [13] B. Kazmierczak and V. Volpert, Travelling calcium waves in systems with non-diffusing buffers,, Mathematical Models and Methods in Applied Sciences, 18 (2008), 883. doi: 10.1142/S0218202508002899. [14] B. Kazmierczak and V. Volpert, Existence of heteroclinic orbits for systems satisfying monotonicity conditions,, Nonlinear Analysis: Theory, 55 (2003), 467. doi: 10.1016/S0362-546X(03)00247-5. [15] J. Keener and J. Sneyd, "Mathematical Physiology,", Springer-Verlag, (1998). [16] J. D. Murray, "Mathematical Biology,", 2nd edition, (1993). doi: 10.1007/b98869. [17] K. J. Palmer, Exponential dichotomies and transversal homoclinic points,, J. Diff. Equat., 20 (1984), 225. doi: 10.1016/0022-0396(84)90082-2. [18] Z. Peradzynski, Diffusion of calcium in biological tissues and accompanying mechano-chemical effects,, Arch. Mech., 62 (2010), 423. [19] Z. Peradzynski and B. Kazmierczak, On mechano-chemical calcium waves,, Archive of Applied Mechanics, 74 (2005), 827. doi: 10.1007/s00419-005-0392-7. [20] K. Piechor, Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment,, Arch. Mech., 64 (2012), 477. [21] M. Sato, N. Ohshima and R. M. Nerem, Viscoelastic properties of cultured porcine aortic endothelial cells exposed to shear stress,, Journal of Biomechanics, 29 (1996), 461. doi: 10.1016/0021-9290(95)00069-0. [22] J. Sneyd, P. D. Dalez and A. Duffy, Traveling waves in buffered systems: Applications to calcium waves,, SIAM J. Appl. Math., 58 (1998), 1178. doi: 10.1137/S0036139996305074. [23] A. E. Taylor, "Introduction to Functional Analysis,", J. Wiley and Sons, (1958). [24] A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems,", AMS, (1994). [25] S. H. Young, H. S. Ennes, J. A. McRoberts, V. V. Chaban, S. K. Dea and E. A. Mayer, Calcium waves in colonic myocytes produced by mechanical and receptor-mediated stimulation,, Am. J. Physiol. Gastrointest. Liver Physiol., 276 (1999), 1204.

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##### References:
 [1] D. Bia, R. Armentano, D. Craiem, J. Grignola, F. Gines, A. Simon and J. Levenson, Smooth muscle role on pulmonary arterial function during acute pulmonary hypertension in sheep,, Acta Physiol. Scand., 181 (2004), 359. doi: 10.1111/j.1365-201X.2004.01294.x. [2] E. C. M. Crooks, On the Vol'pert theory of travelling wave solutions for parabolic equations,, Nonlinear Analysis TM & A, 26 (1996), 1621. doi: 10.1016/0362-546X(95)00038-W. [3] P. B. Dobrin and J. M. Doyle, Vascular smooth muscle and the anisotropy of dog carotid artery,, Circ. Res., 27 (1970), 105. doi: 10.1161/01.RES.27.1.105. [4] A. Doyle, W. Marganski and J. Lee, Calcium transients induce spatially coordinated increases in traction force during the movement of fish keratocytes,, Journal of Cell Science, 117 (2004), 2203. doi: 10.1242/jcs.01087. [5] M. Falcke, Reading the patterns in living cells - the physics of $Ca^{2+}$ signaling,, Advances in Physics, 53 (2004), 255. [6] G. Flores, A. Minzoni, K. Mischaikov and V. Moll, Post-fertilization travelling waves on eggs,, Nonlinear Analysis: Theory, 36 (1999), 45. doi: 10.1016/S0362-546X(97)00696-2. [7] D. Hong, D. Jaron, D. G. Buerk and K. A. Barbee, Heterogeneous response of microvascular endothelial cells to shear stress,, Am. J. Physiol. (Heart Circ. Physiol.), 290 (2006), 2498. doi: 10.1152/ajpheart.00828.2005. [8] L. F. Jaffe, The path of calcium in cytosolic calcium oscillations: A unifying hypothesis,, Proc. Natl. Acad. Sci. USA, 88 (1991), 9883. doi: 10.1073/pnas.88.21.9883. [9] L. F. Jaffe, Stretch-activated calcium channels relay fast calcium waves propagated by calcium-induced calcium influx,, Biol. Cell, 99 (2007), 75. doi: 10.1042/BC20060031. [10] N. R. Jorgensen, S. C. Teilmann, Z. Henriksen, R. Civitelli, O. H. Sorensen and T. H. Steinberg, Activation of L-type calcium channels is required for gap junction-mediated intercellular calcium signaling in osteoblastic cells,, J. Biological Chemistry, 278 (2003), 4082. doi: 10.1074/jbc.M205880200. [11] B. Kazmierczak and Z. Peradzynski, Calcium waves with fast buffers and mechanical effects,, Journal of Mathematical Biology, 62 (2011), 1. doi: 10.1007/s00285-009-0323-2. [12] B. Kazmierczak and V. Volpert, Calcium waves in systems with immobile buffers as a limit of waves for systems with non zero diffusion,, Nonlinearity, 21 (2008), 71. doi: 10.1088/0951-7715/21/1/004. [13] B. Kazmierczak and V. Volpert, Travelling calcium waves in systems with non-diffusing buffers,, Mathematical Models and Methods in Applied Sciences, 18 (2008), 883. doi: 10.1142/S0218202508002899. [14] B. Kazmierczak and V. Volpert, Existence of heteroclinic orbits for systems satisfying monotonicity conditions,, Nonlinear Analysis: Theory, 55 (2003), 467. doi: 10.1016/S0362-546X(03)00247-5. [15] J. Keener and J. Sneyd, "Mathematical Physiology,", Springer-Verlag, (1998). [16] J. D. Murray, "Mathematical Biology,", 2nd edition, (1993). doi: 10.1007/b98869. [17] K. J. Palmer, Exponential dichotomies and transversal homoclinic points,, J. Diff. Equat., 20 (1984), 225. doi: 10.1016/0022-0396(84)90082-2. [18] Z. Peradzynski, Diffusion of calcium in biological tissues and accompanying mechano-chemical effects,, Arch. Mech., 62 (2010), 423. [19] Z. Peradzynski and B. Kazmierczak, On mechano-chemical calcium waves,, Archive of Applied Mechanics, 74 (2005), 827. doi: 10.1007/s00419-005-0392-7. [20] K. Piechor, Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment,, Arch. Mech., 64 (2012), 477. [21] M. Sato, N. Ohshima and R. M. Nerem, Viscoelastic properties of cultured porcine aortic endothelial cells exposed to shear stress,, Journal of Biomechanics, 29 (1996), 461. doi: 10.1016/0021-9290(95)00069-0. [22] J. Sneyd, P. D. Dalez and A. Duffy, Traveling waves in buffered systems: Applications to calcium waves,, SIAM J. Appl. Math., 58 (1998), 1178. doi: 10.1137/S0036139996305074. [23] A. E. Taylor, "Introduction to Functional Analysis,", J. Wiley and Sons, (1958). [24] A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems,", AMS, (1994). [25] S. H. Young, H. S. Ennes, J. A. McRoberts, V. V. Chaban, S. K. Dea and E. A. Mayer, Calcium waves in colonic myocytes produced by mechanical and receptor-mediated stimulation,, Am. J. Physiol. Gastrointest. Liver Physiol., 276 (1999), 1204.
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