Calcium wave propagation by calcium-induced calcium release: An unusual excitable system. (English) Zbl 0758.92007
Summary: We discuss in detail the behaviour of a model, proposed by A. Goldbeter, G. Dupont and M. J. Berridge [Proc. Natl. Acad. Sci. 87, 1461-1465 (1990)], for intracellular calcium wave propagation by calcium-induced calcium release, focusing our attention on excitability and the propagation of waves in one spatial dimension. The model with no diffusion behaves like a generic excitable system, and threshold behaviour, excitability and oscillations can be understood within this general framework.
However, when diffusion is included, the model no longer behaves like a generic excitable system; the fast and slow variables are not distinct and previous results on excitable systems do not necessarily apply. We consider a piecewise linear simplification of the model, and construct travelling pulse and periodic plane wave solutions to the simplified model. The analogous behaviour in the full model is studied numerically. Goldbeter’s model for calcium-induced calcium release is an excitable system of a type not previously studied in detail.
However, when diffusion is included, the model no longer behaves like a generic excitable system; the fast and slow variables are not distinct and previous results on excitable systems do not necessarily apply. We consider a piecewise linear simplification of the model, and construct travelling pulse and periodic plane wave solutions to the simplified model. The analogous behaviour in the full model is studied numerically. Goldbeter’s model for calcium-induced calcium release is an excitable system of a type not previously studied in detail.
MSC:
| 92C30 | Physiology (general) |
Keywords:
travelling pulse solutions; intracellular calcium wave propagation; calcium-induced calcium release; one spatial dimension; threshold behaviour; excitability; oscillations; piecewise linear simplification; periodic plane wave solutionsSoftware:
AUTO-86References:
| [1] | Backx, P. H., P. P. de Tombe, J. H. K. Van Deen, B. J. M. Mulder and H. E. D. J. ter Keurs. 1989. A model of propagating calcium-induced calcium release mediated by calcium diffusion.J. gen. Physiol. 93, 963–977. · doi:10.1085/jgp.93.5.963 |
| [2] | Berridge, M. J. 1989. Cell signalling through cytoplasmic calcium oscillations. InCell to Cell Signalling: From Experiment to Theoretical Models, A. Goldbeter (Ed.), pp. 449–459. London: Academic Press. |
| [3] | Berridge, M. J. 1990. Calcium oscillations.J. Biol. Chem. 265, 9583–9586. |
| [4] | Berridge, M. J. 1991. Cytoplasmic calcium oscillations: a two pool model.Cell Calcium 12, 63–72. · doi:10.1016/0143-4160(91)90009-4 |
| [5] | Berridge, J. J. and A. Galione. 1988. Cytosolic calcium oscillators.FASEB J. 2, 3074–3082. |
| [6] | Berridge, M. J. and R. F. Irvine. 1989. Inositol phosphates and cell signalling.Nature 341, 197–205. · doi:10.1038/341197a0 |
| [7] | Bezprozvanny, I., J. Watras and B. E. Ehrlich. 1991. Bell-shaped calcium-response curves of Ins(1,4,5)P3-and calcium-gated channels from endoplasmic reticulum of cerebellum.Nature 351, 751–754. · doi:10.1038/351751a0 |
| [8] | Britton, N. F. 1986.Reaction-Diffusion Equations and Their Applications to Biology, London: Academic Press. · Zbl 0602.92001 |
| [9] | Busa, W. B. and R. Nuccitelli. 1985. An elevated free cytosolic Ca2+ wave follows fertilization in eggs of the frog,Xenopus laevis.J. Cell Biol. 100, 1325–1329. · doi:10.1083/jcb.100.4.1325 |
| [10] | Cannell, M. B. and D. G. Allen. 1984. Model of calcium movements during activation in the sarcomere of frog skeletal muscle,Biophys. J. 45, 913–925. · doi:10.1016/S0006-3495(84)84238-1 |
| [11] | Casten, R. G., H. Cohen and P. A. Lagerstrom. 1975. Perturbation analysis of an approximation to the Hodgkin-Huxley theory.Q. Appl. Math. 32, 365–402. · Zbl 0365.92005 |
| [12] | Charles, A. C., J. E. Merrill, E. R. Dirksen and M. J. Sanderson. 1991. Intercellular signaling in glial cells: calcium waves and oscillations in response to mechanical stimulation and glutamate.Neuron. 6, 983–992. · doi:10.1016/0896-6273(91)90238-U |
| [13] | Cheer, A., J-P. Vincent, R. Nuccitelli and G. Oster. 1987. Cortical activity in vertebrate eggs I: the activation waves.J. theor. Biol. 124, 377–404. · doi:10.1016/S0022-5193(87)80217-5 |
| [14] | Cobbold, P. H., A. Sanchez-Bueno and C. J. Dixon. 1991. The hepatocyte calcium oscillator.Cell Calcium. 12, 87–95. · doi:10.1016/0143-4160(91)90011-3 |
| [15] | Cohen, D. S., J. C. Neu and R. R. Rosales. 1978. Rotating spiral wave solutions of reaction-diffusion equations.SIAM J. appl. Math. 35, 536–547. · Zbl 0395.35075 · doi:10.1137/0135045 |
| [16] | Cuthbertson, K. S. R. 1989. Intracellular calcium oscillators, InCell to Cell Signaling: From Experiments to Theoretical Models, A. Goldbeter (Ed.) pp. 435–447. London: Academic Press. |
| [17] | Cuthbertson, K. S. R. and T. R. Chay. 1991. Modelling receptor-controlled intracellular calcium oscillators.Cell Calcium. 12, 97–109. · doi:10.1016/0143-4160(91)90012-4 |
| [18] | Dockery, J. D. and J. P. Keener. 1989. Diffusive effects on dispersion in excitable media.SIAM J. app. Math. 49, 539–566. · Zbl 0703.35091 · doi:10.1137/0149031 |
| [19] | Doedel, E. 1986.Software for continuation and bifurcation problems in ordinary differential equations. California Institute of Technology. |
| [20] | Duffy, M. R., N. F. Britton and J. D. Murray. 1980. Spiral wave solutions of practical reaction-diffusion systems.SIAM J. appl. Math. 39, 8–13. · Zbl 0451.35075 · doi:10.1137/0139002 |
| [21] | Dupont, G. and A. Goldbeter. 1989. Theoretical insights into the origin of signal-induced calcium oscillations, InCell to Cell Signalling: From Experiments to Theoretical Models, A. Goldbeter (Ed.), pp. 461–474. London: Academic Press. |
| [22] | Dupont, G., M. J. Berridge and A. Goldbeter. 1991. Signal-induced Ca2+ oscillations: properties of a model based on Ca2+-induced Ca2+ release.Cell Calcium.12, 73–85. · doi:10.1016/0143-4160(91)90010-C |
| [23] | Endo, M., M. Tanaka and Y. Ogawa. 1970. Calcium induced release of calcium from the sarcoplasmic reticulum of skinned skeletal muscle fibres.Nature 228, 34–36. · doi:10.1038/228034a0 |
| [24] | Fabiato, A. 1983. Calcium-induced release of calcium from the cardiac sarcoplasmic reticulum.Am. J. Physiol. 245, C1-C14. |
| [25] | Fabiato, A. and F. Fabiato. 1975. Contractions induced by a calcium-triggered release of calcium from the sarcoplasmic reticulum of single skinned cardiac cells.J. Physiol. 249, 469–495. |
| [26] | Finch, E. A., T. J. Turner and S. M. Goldin. 1991. Calcium as a coagonist of inositol 1,4,5-trisphosphate-induced calcium release.Science 252, 443–446. · doi:10.1126/science.2017683 |
| [27] | FitzHugh, R. 1961. Impulses and physiological states in theoretical models of nerve membrane.Biophys. J. 1, 445–466. · doi:10.1016/S0006-3495(61)86902-6 |
| [28] | FitzHugh, R. 1969. Impulses and physiological states in models of nerve membrane. InBiological Engineering, H. P. Schwan (Ed.), pp. 1–85. New York: McGraw-Hill. |
| [29] | Gilkey, J. C., L. F. Jaffe, E. B. Ridgway and G. T. Reynolds. 1978. A free calcium wave traverses the activating egg of the medakaoryzias latipes.J. Cell Biol. 76, 448–466. · doi:10.1083/jcb.76.2.448 |
| [30] | Girard, S., A. Luckhoff, J. Lechleiter, J. Sneyd and D. Clapham. 1992. Two-dimensional model of calcium waves reproduces the patterns observed inXenopus oocytes.Biophys. J. 61, 509–517. · doi:10.1016/S0006-3495(92)81855-6 |
| [31] | Goldbeter, A., G. Dupont and M. J. Berridge. 1990. Minimal model for signal-induced Ca2+ oscillations and for their frequency encoding through protein phosphorylation,Proc. natn. Acad. Sci. 87, 1461–1465. · doi:10.1073/pnas.87.4.1461 |
| [32] | Hagan, P. S. 1982. Spiral waves in reaction-diffusion equations.SIAM J. appl. Math. 42, 762–786. · Zbl 0507.35007 · doi:10.1137/0142054 |
| [33] | Harootunian, A. T., J. P. Y. Kao, S. Paranjape, S. R. Adams, B. V. L. Potter, and R. Y. Tsien. 1991. Cytosolic Ca2+ oscillations in REF52 fibroblasts: Ca2+-stimulated IP3 production or voltage-dependent Ca2+ channels as key positive feedback elements.Cell Calcium 12, 153–164. · doi:10.1016/0143-4160(91)90017-9 |
| [34] | Hastings, S. P. 1974. The existence of periodic solutions to Nagumo’s equation.Q. J. Math. 25, 369–378. · Zbl 0305.34058 · doi:10.1093/qmath/25.1.369 |
| [35] | Hastings, S. P. 1976. On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations.Q. J. Math. 27, 123–134. · Zbl 0322.92008 · doi:10.1093/qmath/27.1.123 |
| [36] | Iino, M. 1990. Biphasic Ca2+ dependence of inositol 1,4,5-trisphosphate-induced Ca2+ release in smooth muscle cells of the Guinea PigTaenia Caeci.J. gen. Physiol. 95, 1103–1122. · doi:10.1085/jgp.95.6.1103 |
| [37] | Jacob, R. 1991. Calcium oscillations in endothelial cells.Cell Calcium 12, 127–134. · doi:10.1016/0143-4160(91)90014-6 |
| [38] | Keener, J. P. 1980. Waves in excitable media.SIAM J. appl. Math. 39, 528–548. · Zbl 0457.35082 · doi:10.1137/0139043 |
| [39] | Keener, J. P. 1986. A geometrical theory for spiral waves in excitable media.SIAM J. appl. Math. 46, 1039–1056. · Zbl 0655.35046 · doi:10.1137/0146062 |
| [40] | Kopell, N. and L. N. Howard. 1973. Plane wave solutions to reaction-diffusion equations.Stud. appl. Math. 52, 291–328. · Zbl 0305.35081 |
| [41] | Kopell, N. and L. N. Howard. 1981. Target pattern and spiral solutions to reaction-diffusion equations with more than one space dimension.Adv. Appl. Math. 2, 417–449. · Zbl 0486.35011 · doi:10.1016/0196-8858(81)90043-9 |
| [42] | Kuba, K. and S. Takeshita. 1981. Simulation of intracellular Ca2+ oscillation in a sympathetic neurone.J. theor. Biol. 93, 1009–1031. · doi:10.1016/0022-5193(81)90352-0 |
| [43] | Lane, D. C., J. D. Murray and V. S. Manoranjan. 1987. Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs.IMA J. Math. appl. Med. Biol. 4, 309–331. · Zbl 0623.92008 · doi:10.1093/imammb/4.4.309 |
| [44] | Lechleiter, J., S. Girard, D. Clapham and E. Peralta. 1991a. Subcellular patterns of calcium release determined by G protein-specific residues of muscarinic receptors.Nature 350, 505–508. · doi:10.1038/350505a0 |
| [45] | Lechleiter, J., S. Girard, E. Peralta and D. Clapham. 1991b. Spiral calcium wave propagation and annihilation inXenopus laevis oocytes.Science 252, 123–126. · doi:10.1126/science.2011747 |
| [46] | McKean, H. P. 1970. Nagumo’s equation.Adv. Math. 4, 209–223. · Zbl 0202.16203 · doi:10.1016/0001-8708(70)90023-X |
| [47] | Maginu, K. 1985. Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction-diffusion equations.SIAM J. appl. Math. 45, 750–774. · Zbl 0602.35052 · doi:10.1137/0145044 |
| [48] | Meyer, T. and L. Stryer. 1988. Molecular model for receptor-stimulated calcium spiking.Proc. natn. Acad. Sci. 85, 5051–5055. · doi:10.1073/pnas.85.14.5051 |
| [49] | Murray, J. D. 1989.Mathematical Biology. Berlin: Springer-Verlag. · Zbl 0682.92001 |
| [50] | Murray, J. D. and G. F. Oster. 1984. Generation of biological pattern and form.IMA J. Math. appl. Med. Biol. 1, 51–75. · Zbl 0611.92003 · doi:10.1093/imammb/1.1.51 |
| [51] | Neu, J. C. 1979. Chemical waves and the diffusive coupling of limit cycle oscillators.SIAM J. appl Math. 36, 509–515. · Zbl 0418.35013 · doi:10.1137/0136038 |
| [52] | Oster, G. F. and G. M. Odell. 1984. Mechanics of cytogels I: oscillations in Physarum.Cell Motility 4, 469–503. · doi:10.1002/cm.970040606 |
| [53] | Parker, I. and I. Ivorra. 1990. Inhibition by Ca2+ of inositol trisphosphate-mediated Ca2+ liberation: a possible mechanism for oscillatory release of Ca2+.Proc. natn. Acad. Sci. 87, 260–264. · doi:10.1073/pnas.87.1.260 |
| [54] | Peskin, C. S. 1976.Partial Differential Equations in Biology. New York: Courant Insitute of Mathematical Sciences, New York University. · Zbl 0329.35001 |
| [55] | Petersen, O. H., D. V. Gallacher, M. Wakui, D. I. Yule, C. C. H. Petersen and E. C. Toescu. 1991. Receptor-activated cytoplasmic Ca2+ oscillations in pancreatic acinar cells: generation and spreading of Ca2+ signals.Cell Calcium 12, 135–144. · doi:10.1016/0143-4160(91)90015-7 |
| [56] | Rinzel, J. 1976. Simple model equations for active nerve conduction and passive neuronal integration.Lect. Math. Life Sci. 8, 125–164. · Zbl 0351.92009 |
| [57] | Rinzel, J. and J. B. Keller. 1973. Traveling wave solutions of a nerve conduction equation.Biophys J. 13, 1313–1337. · doi:10.1016/S0006-3495(73)86065-5 |
| [58] | Sanderson, M. J., A. C. Charles and E. R. Dirksen. 1990. Mechanical stimulation and intercellular communication increases intracellular Ca2+ in epithelial cells.Cell Regulat. 1, 585–596. |
| [59] | Sauvé, R., A. Diarra, M. Chahine, C. Simoneau, N. Morier and G. Roy. 1991. Ca2+ oscillations induced by histamine H1 receptor stimulation in HeLa cells: Fura-2 and patch clamp analysis.Cell Calcium. 12, 165–176. · doi:10.1016/0143-4160(91)90018-A |
| [60] | Swillens, S. and D. Mercan. 1990. Computer simulation of a cytosolic calcium oscillator.Biochem. J. 271, 835–838. |
| [61] | Thomas, A. P., D. C. Renard and T. A. Rooney. 1991. Spatial and temporal organization of calcium signalling in hepatocytes.Cell Calcium 12, 111–126. · doi:10.1016/0143-4160(91)90013-5 |
| [62] | Troy, W. C. 1976. Bifurcation phenomena in FitzHugh’s nerve conduction equations.J. Math. analys. appl. 54, 678–690. · Zbl 0329.92001 · doi:10.1016/0022-247X(76)90187-6 |
| [63] | Tsien, R. W. and R. Y. Tsien. 1990. Calcium channels, stores, and oscillations,A. Rev. Cell Biol. 6, 715–760. · doi:10.1146/annurev.cb.06.110190.003435 |
| [64] | Tsunoda, Y. 1991. Oscillatory Ca2+ signaling and its cellular function.The New Biol. 3, 3–17. |
| [65] | Zykov, V. S. 1980. Analytic evaluation of the dependence of the speed of excitation wave in a twodimensional excitable medium on the curvature of its front.Biophys. 25, 906–911. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
