Abstract
We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca2+) in living cells. Modeling Ca2+ release as a threshold process allows the explicit construction of traveling wave solutions to probe the dependence of Ca2+ wave speed on physiologically important parameters such as the threshold for Ca2+ release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca2+ resequestration from the cytosol to the ER, and the total [Ca2+] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca2+ waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called ‘tango waves’ and the dependence of Ca2+ wave speed on the total [Ca2+].
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References
Berridge M.J. (1993). Inositol trisphosphate and calcium signalling. Nature 361(6410): 315–325
Whitaker M. (2006). Calcium at fertilization and in early development. Physiol. Rev. 86(1): 25–88
Miyazaki S., Ito M. (2006). Calcium signals for egg activation in mammals. J. Pharmacol. Sci. 100(5): 545–552
Bers D.M. (2002). Cardiac excitation–contraction coupling. Nature 415(6868): 198–205
Clapham D.E. (1995). Calcium signaling. Cell 80(2): 259–268
Keizer J., Li Y.X., Stojilkovic S., Rinzel J. (1995). InsP3-induced Ca2+ excitability of the endoplasmic reticulum. Mol. Biol. Cell 6(8): 945–951
Ghosh A., Greenberg M.E. (1995). Calcium signaling in neurons: molecular mechanisms and cellular consequences. Science 268(5208): 239–247
Berridge M.J. (1997). Elementary and global aspects of calcium signalling. J. Physiol. 499(Pt 2): 291–306
Berridge M.J. (1998). Neuronal calcium signaling. Neuron 21(1): 13–26
Bezprozvanny I., Ehrlich B.E. (1995). The inositol 1,4,5-trisphosphate (InsP3) receptor. J. Membr. Biol. 145(3): 205–216
Ehrlich B.E. (1995). Functional properties of intracellular calcium-release channels. Curr. Opin. Neurobiol. 5(3): 304–209
Li Y.X., Keizer J., Stojilkovic S.S., Rinzel J. (1995). Ca2+ excitability of the ER membrane: an explanation for IP3-induced Ca2+ oscillations. Am. J. Physiol. 269(5 Pt 1): C1079–C1092
Lechleiter J.D., Clapham D.E. (1992). Molecular mechanisms of intracellular calcium excitability in X. laevis oocytes. Cell 69(2): 283–294
Ridgway E.B., Gilkey J.C., Jaffe L.F. (1977). Free calcium increases explosively in activating medaka eggs. Proc. Natl. Acad. Sci. USA 74(2): 623–627
Nuccitelli R., Yim D.L., Smart T. (1993). The sperm-induced Ca2+ wave following fertilization of the Xenopus eggs requires the production of Ins(1,4,5)P3. Dev. Biol. 158: 200–212
Stricker S.A., Centonze V.E., Melendez R.F. (1994). Calcium dynamics during starfish oocyte maturation and fertilization. Dev. Biol. 166(1): 34–58
Dumollard R., McDougall A., Rouvire C., Sardet C. (2004). Fertilisation calcium signals in the ascidian egg. Biol. Cell 96(1): 29–36
Cheng H., Lederer M.R., Lederer W.J., Cannell M.B. (1996). Calcium sparks and \([\mathrm{Ca}^{2+}]_i\) waves in cardiac myocytes. Am. J. Physiol. 270(1): C148–C159
Lee C.H., Poburko D., Kuo K.H., Seow C.Y., van Breemen C. (2002). Ca2+ oscillations, gradients, and homeostasis in vascular smooth muscle. Am. J. Physiol. Heart Circ. Physiol. 282(5): H1571–H1583
Ying X., Minamiya Y., Fu C., Bhattacharya J. (1996). Ca2+ waves in lung capillary endothelium. Circ. Res. 79(4): 898–908
Fink C.C., Slepchenko B., Moraru I.I., Watras J., Schaff J.C., Loew L.M. (2000). An image-based model of calcium waves in differentiated neuroblastoma cells. Biophys. J. 79(1): 163–83
Fiacco T.A., McCarthy K.D. (2006). Astrocyte calcium elevations: properties, propagation and effects on brain signaling. Glia 54(7): 676–690
Kiselyov K., Wang X., Shin D.M., Zang W., Muallem S. (2006). Calcium signaling complexes in microdomains of polarized secretory cells. Cell Calcium 40(5–6): 451–459
Dupont G., Goldbeter A. (1992). Oscillations and waves of cytosolic calcium: insights from theoretical models. Bioessays 14(7): 485–493
Atri A., Amundson J., Clapham D., Sneyd J. (1993). A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte. Biophys. J. 65(4): 1727–1739
Sneyd J., Girard S., Clapham D. (1993). Calcium wave propagation by calcium-induced calcium release: an unusual excitable system. Bull. Math. Biol. 55(2): 315–344
Dupont G., Goldbeter A. (1994). Properties of intracellular Ca2+ waves generated by a model based on Ca2+-induced Ca2+ release. Biophys. J. 67(6): 2191–2204
Jafri M.S., Keizer J. (1994). Diffusion of inositol 1,4,5-trisphosphate but not Ca2+ is necessary for a class of inositol 1,4,5-trisphosphate-induced Ca2+ waves. Proc. Natl. Acad. Sci. USA 91(20): 9485–9489
Jafri M.S., Keizer J. (1995). On the roles of Ca2+ diffusion, Ca2+ buffers and the endoplasmic reticulum in IP3-induced Ca2+ waves. Biophys. J. 69(5): 2139–2153
Jafri S.M., Keizer J. (1997). Agonist-induced calcium waves in oscillatory cells: a biological example of burgers’ equation. Bull. Math. Biol. 59(6): 1125–1144
Wagner J., Li Y.X., Pearson J., Keizer J. (1998). Simulation of the fertilization Ca2+ wave in Xenopus laevis eggs. Biophys. J. 75(4): 2088–2097
Sneyd, J.: An introduction to Mathematical Modeling in Physiology, Cell Biology, and Immunology chapter Calcium excitability. American Mathematical Society, pp 83–118 (2002)
Fall C.P., Wagner J.M., Loew L.M., Nuccitelli R. (2004). Cortically restricted production of IP3 leads to propagation of the fertilization Ca2+ wave along the cell surface in a model of the Xenopus egg. J. Theor. Biol. 231(4): 487–496
Falcke M., Li Y., Lechleiter D.J., Camacho P. (2003). Modeling the dependence of the period of intracellular Ca2+ waves on SERCA expression. Biophys. J. 85: 1474–1481
De Young G.W., Keizer J. (1992). A single pool IP3-receptor based model for agonist stimulated Ca2+ oscillations. Proc. Natl. Acad. Sci. USA 89: 9895–9899
Li Y.X., Rinzel J. (1994). Equations for InsP3 receptor-mediated \([\mathrm{Ca}^{2+}]_i\) oscillations derived from a detailed kinetic model: a Hodgkin–Huxley like formalism. J. Theor. Biol. 166(4): 461–473
Keener, J., Sneyd, J.: Mathematical Physiology. Springer, Heidelberg (1998)
Smith, G.D., Pearson, J.E., Keizer, J.E.: Modeling intracellular Ca2+ waves and sparks. In: Fall C.P., Marland E.S., Wagner J.M., Tyson J.J. (eds.) Computational Cell Biology, pp 198–229. Springer, Heidelberg (2002)
Sneyd J., Keizer J., Sanderson M.J. (1995). Mechanisms of calcium oscillations and waves: a quantitative analysis. FASEB J. 9(14): 1463–1472
Li Y.-X. (2003). Tango waves in a bidomain model of fertilization calcium waves. Physica D 186: 27–49
Stricker S.A. (1996). Repetitive calcium waves induced by fertilization in the nemertean worm Cerebratulus lacteus. Dev. Biol. 176: 243–263
Yoshida M., Sensui N., Inoue T., Morisawa M., Mikoshiba K. (1998). Role of two series of Ca2+ oscillations in activation of ascidian eggs. Dev. Biol. 203: 122–133
Prat A., Li Y.-X. (2003). Stability of front solutions in inhomogeneous media. Physica D 186: 50–68
Yao Y., Choi J., Parker I. (1995). Quantal puffs of intracellular Ca2+ evoked by inositol trisphosphate in Xenopus oocytes. J. Physiol. 482: 533–553
Parker I., Choi J., Yao Y. (1996). Elementary events of InsP3-induced Ca2+ liberation in Xenopus oocytes: hot spots, puffs and blips. Cell Calcium 20(2): 105–21
Bugrim A.E., Zhabotinsky A.M., Epstein I.R. (1997). Calcium waves in a model with a random spatially discrete distribution of Ca2+ release sites. Biophys. J. 73(6): 2897–906
Cheng H., Lederer J.W., Cannell M.B. (1993). Calcium sparks: elementary events underlying excitation–contraction coupling in heart muscle. Science 262(5134): 740–4
Keizer J.E., Smith G.D. (1998). Spark-to-wave transition: saltatory transmission of calcium waves in cardiac myocytes. Biophys. Chem. 72: 87–100
Keizer J., Smith G.D., Ponce-Dawson S., Pearson J.E. (1998). Saltatory propagation of Ca2+ waves by Ca2+ sparks. Biophys. J. 75(2): 595–600
Pearson J.E., Ponce Dawson S. (1998). Crisis on skid row. Physica A 257: 141–148
Dawson S.P., Keizer J., Pearson J.E. (1999). Fire-diffuse-fire model of dynamics of intracellular calcium waves. Proc. Natl. Acad. Sci. USA 96: 6060–6063
Coombes S. (2001). The effect of ion pumps on the speed of travelling waves in the fire-diffuse-fire model of Ca2+ release. Bull. Math. Biol. 63: 1–20
Coombes S., Timofeeva Y. (2003). Sparks and waves in a stochastic fire-diffuse-fire model of Ca2+ release. Phys. Rev. E 68: 021915
Coombes S., Hinch R., Timofeeva Y. (2004). Receptors, sparks and waves in a fire-diffuse-fire framework for calcium release. Progress Biophys. Mol. Biol. 85: 197–216
Timofeeva Y., Coombes S. (2004). Directed percolation in a two-dimensional stochastic fire-diffuse-fire model. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(6 Pt 1): 062901
Timofeeva Y., Coombes S. (2003). Wave bifurcation and propagation failure in a model of calcium release. J. Math. Biol. 47: 249–269
Shannon T.R., Wang F., Puglisi J., Weber C., Bers D.M. (2004). A mathematical treatment of integrated Ca dynamics within the ventricular myocyte. Biophys. J. 87: 3351–3371
Bressloff P.C., Folias S.E., Prat A., Li X.Y. (2003). Oscillatory waves in inhomogeneous neural media. Phys. Rev. Lett. 91(17): 178101
Prat A., Li X.Y., Bressloff P. (2005). Inhomogeneity-induced bifurcation of stationary and oscillatory pulses. Physica D 202: 177–99
Terentyev D., Viatchenko-Karpinski S., Valdivia H.H., Escobar A.L., Gyorke S. (2002). Luminal Ca controls termination and refractory behaviour of Ca induced Ca release in cardiac myocytes. Circ. Res. 91: 414–420
Keller M., Kao J.P., Egger M., Niggli E. (2007). Calcium waves driven by “sensitization” wave-fronts. Cardiovascular Res. 74(1): 39–45
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Thul, R., Smith, G.D. & Coombes, S. A bidomain threshold model of propagating calcium waves. J. Math. Biol. 56, 435–463 (2008). https://doi.org/10.1007/s00285-007-0123-5
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DOI: https://doi.org/10.1007/s00285-007-0123-5