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

Calcium waves with mechano-chemical couplings

Bogdan Kazmierczak 1, and  Zbigniew Peradzynski 2,

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.
Keywords: mechanochemical coupling., reaction-diffusion systems, Calcium waves.
Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 35Q9.
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), 359-366. doi: 10.1111/j.1365-201X.2004.01294.x.  Google Scholar

[2]

E. C. M. Crooks, On the Vol'pert theory of travelling wave solutions for parabolic equations, Nonlinear Analysis TM & A, 26 (1996), 1621-1642. doi: 10.1016/0362-546X(95)00038-W.  Google Scholar

[3]

P. B. Dobrin and J. M. Doyle, Vascular smooth muscle and the anisotropy of dog carotid artery, Circ. Res., 27 (1970), 105-119. doi: 10.1161/01.RES.27.1.105.  Google Scholar

[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), 2203-2214. doi: 10.1242/jcs.01087.  Google Scholar

[5]

M. Falcke, Reading the patterns in living cells - the physics of $Ca^{2+}$ signaling, Advances in Physics, 53 (2004), 255-440. Google Scholar

[6]

G. Flores, A. Minzoni, K. Mischaikov and V. Moll, Post-fertilization travelling waves on eggs, Nonlinear Analysis: Theory, Methods and Applications, 36 (1999), 45-62. doi: 10.1016/S0362-546X(97)00696-2.  Google Scholar

[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), 2498-2508. doi: 10.1152/ajpheart.00828.2005.  Google Scholar

[8]

L. F. Jaffe, The path of calcium in cytosolic calcium oscillations: A unifying hypothesis, Proc. Natl. Acad. Sci. USA, 88 (1991), 9883-9887. doi: 10.1073/pnas.88.21.9883.  Google Scholar

[9]

L. F. Jaffe, Stretch-activated calcium channels relay fast calcium waves propagated by calcium-induced calcium influx, Biol. Cell, 99 (2007), 75-184. doi: 10.1042/BC20060031.  Google Scholar

[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), 4082-4086. doi: 10.1074/jbc.M205880200.  Google Scholar

[11]

B. Kazmierczak and Z. Peradzynski, Calcium waves with fast buffers and mechanical effects, Journal of Mathematical Biology, 62 (2011), 1-38. doi: 10.1007/s00285-009-0323-2.  Google Scholar

[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), 71-96. doi: 10.1088/0951-7715/21/1/004.  Google Scholar

[13]

B. Kazmierczak and V. Volpert, Travelling calcium waves in systems with non-diffusing buffers, Mathematical Models and Methods in Applied Sciences, 18 (2008), 883-912 doi: 10.1142/S0218202508002899.  Google Scholar

[14]

B. Kazmierczak and V. Volpert, Existence of heteroclinic orbits for systems satisfying monotonicity conditions, Nonlinear Analysis: Theory, Methods and Applications, 55 (2003), 467-491. doi: 10.1016/S0362-546X(03)00247-5.  Google Scholar

[15]

J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, 1998.  Google Scholar

[16]

J. D. Murray, "Mathematical Biology," 2nd edition, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar

[17]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Equat., 20 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[18]

Z. Peradzynski, Diffusion of calcium in biological tissues and accompanying mechano-chemical effects, Arch. Mech., 62 (2010), 423-440.  Google Scholar

[19]

Z. Peradzynski and B. Kazmierczak, On mechano-chemical calcium waves, Archive of Applied Mechanics, 74 (2005), 827-833. doi: 10.1007/s00419-005-0392-7.  Google Scholar

[20]

K. Piechor, Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment, Arch. Mech., 64 (2012), pp. 477-509 Google Scholar

[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), 461-467. doi: 10.1016/0021-9290(95)00069-0.  Google Scholar

[22]

J. Sneyd, P. D. Dalez and A. Duffy, Traveling waves in buffered systems: Applications to calcium waves, SIAM J. Appl. Math., 58 (1998), 1178-1192. doi: 10.1137/S0036139996305074.  Google Scholar

[23]

A. E. Taylor, "Introduction to Functional Analysis," J. Wiley and Sons, New York, 1958.  Google Scholar

[24]

A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," AMS, Providence 1994. Google Scholar

[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), 1204-1212. Google Scholar

show all references

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), 359-366. doi: 10.1111/j.1365-201X.2004.01294.x.  Google Scholar

[2]

E. C. M. Crooks, On the Vol'pert theory of travelling wave solutions for parabolic equations, Nonlinear Analysis TM & A, 26 (1996), 1621-1642. doi: 10.1016/0362-546X(95)00038-W.  Google Scholar

[3]

P. B. Dobrin and J. M. Doyle, Vascular smooth muscle and the anisotropy of dog carotid artery, Circ. Res., 27 (1970), 105-119. doi: 10.1161/01.RES.27.1.105.  Google Scholar

[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), 2203-2214. doi: 10.1242/jcs.01087.  Google Scholar

[5]

M. Falcke, Reading the patterns in living cells - the physics of $Ca^{2+}$ signaling, Advances in Physics, 53 (2004), 255-440. Google Scholar

[6]

G. Flores, A. Minzoni, K. Mischaikov and V. Moll, Post-fertilization travelling waves on eggs, Nonlinear Analysis: Theory, Methods and Applications, 36 (1999), 45-62. doi: 10.1016/S0362-546X(97)00696-2.  Google Scholar

[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), 2498-2508. doi: 10.1152/ajpheart.00828.2005.  Google Scholar

[8]

L. F. Jaffe, The path of calcium in cytosolic calcium oscillations: A unifying hypothesis, Proc. Natl. Acad. Sci. USA, 88 (1991), 9883-9887. doi: 10.1073/pnas.88.21.9883.  Google Scholar

[9]

L. F. Jaffe, Stretch-activated calcium channels relay fast calcium waves propagated by calcium-induced calcium influx, Biol. Cell, 99 (2007), 75-184. doi: 10.1042/BC20060031.  Google Scholar

[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), 4082-4086. doi: 10.1074/jbc.M205880200.  Google Scholar

[11]

B. Kazmierczak and Z. Peradzynski, Calcium waves with fast buffers and mechanical effects, Journal of Mathematical Biology, 62 (2011), 1-38. doi: 10.1007/s00285-009-0323-2.  Google Scholar

[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), 71-96. doi: 10.1088/0951-7715/21/1/004.  Google Scholar

[13]

B. Kazmierczak and V. Volpert, Travelling calcium waves in systems with non-diffusing buffers, Mathematical Models and Methods in Applied Sciences, 18 (2008), 883-912 doi: 10.1142/S0218202508002899.  Google Scholar

[14]

B. Kazmierczak and V. Volpert, Existence of heteroclinic orbits for systems satisfying monotonicity conditions, Nonlinear Analysis: Theory, Methods and Applications, 55 (2003), 467-491. doi: 10.1016/S0362-546X(03)00247-5.  Google Scholar

[15]

J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, 1998.  Google Scholar

[16]

J. D. Murray, "Mathematical Biology," 2nd edition, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar

[17]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Equat., 20 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[18]

Z. Peradzynski, Diffusion of calcium in biological tissues and accompanying mechano-chemical effects, Arch. Mech., 62 (2010), 423-440.  Google Scholar

[19]

Z. Peradzynski and B. Kazmierczak, On mechano-chemical calcium waves, Archive of Applied Mechanics, 74 (2005), 827-833. doi: 10.1007/s00419-005-0392-7.  Google Scholar

[20]

K. Piechor, Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment, Arch. Mech., 64 (2012), pp. 477-509 Google Scholar

[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), 461-467. doi: 10.1016/0021-9290(95)00069-0.  Google Scholar

[22]

J. Sneyd, P. D. Dalez and A. Duffy, Traveling waves in buffered systems: Applications to calcium waves, SIAM J. Appl. Math., 58 (1998), 1178-1192. doi: 10.1137/S0036139996305074.  Google Scholar

[23]

A. E. Taylor, "Introduction to Functional Analysis," J. Wiley and Sons, New York, 1958.  Google Scholar

[24]

A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," AMS, Providence 1994. Google Scholar

[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), 1204-1212. Google Scholar

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