On the role of stochastic channel behavior in intracellular Ca(2+) dynamics

ABSTRACT I present a stochastic model for intracellular Ca^sup 2+^ oscillations. The model starts from stochastic binding and dissociation of Ca^sup 2+^ to binding sites on a single subunit of the IP3-receptor channel but is capable of simulating large numbers of clusters for many oscillation periods too. I find oscillations with variable periods ranging from 17 s to 120 s and a standard deviation well in the experimentally observed range. Long period oscillations can be perceived as nucleation phenomenon and can be observed for a large variety of single channel dynamics. Their period depends on the geometric characteristics of the cluster array. Short periods are in the range of the time scale of channel dynamics. Both long and short period oscillations occur for parameters with a nonoscillatory deterministic regime.

INTRODUCTION

The beauty of intracellular Ca^sup 2+^ dynamics is that it allows for observation of the build up of periodic global events from local stochastic events. A theoretical analysis on how the stochasticity of the elemental events shows up in the global events is the subject of this article.

Changes in the cytosolic free Ca^sup 2+^ concentration are used by many cells for signaling (Tsien and Tsien, 1990; Berridge et al., 1998). The rise of that concentration is accomplished beside influx through the plasma membrane by release of Ca^sup 2+^ from intracellular stores like the endoplasmic reticulum (ER). Opening and closing of Ca^sup 2+^ channels on the ER membrane controls the release. Channels are closely packed into clusters (Sun et al., 1998; Thomas et al., 1998; Mak et al., 2000; Mak and Foskett, 1998). The clusters are randomly distributed on the ER membrane. Areas with high cluster density are called focal sites (Lechleiter et al., 1991; Callamaras and Parker, 2000; Marchant and Parker, 2001). Channels open and close stochastically. Stochastic behavior manifests itself as spontaneous release events through single channels or several channels in a cluster (Sun et al., 1998; Callamaras and Parker, 2000; Marchant and Parker, 2001; Thomas et al., 1999; Bootman et al., 1997a).

The solution of that integral equation depends on the configuration of open channels N^sub o^(t). That can be understood by imagining the extreme case of redistributing a large fraction of free Ca^sup 2+^ to immobile buffers by release. The release would decrease the value of A^sup 0^ because immobile buffers do not contribute to it.

The system of Eqs. 8 and 9 was solved every time No(t) changed for the simulations shown in Figs. 2 and 3. 1 used the single cluster profile approach for fast localized variables in all other simulations. The single cluster solutions were determined by solving Eq. 8 with a single cluster site centered at r = 0. The value of A^sup 0^ was determined by Eq. 9 for the concentration fields on the whole area.

I solved all partial differential equations by a multigrid method according to Press et al. (1992). I used a spatial discretization of 4 nm to calculate single cluster profiles. The time step size in stochastic simulations was chosen so that the probability for a transition in a cluster was ~1% and hence for all clusters within an interaction radius ~7%. Because only about one quarter of the transitions changes the open state of channels that yields ~2% probability per time step for a transition changing rates of other transitions within the interaction radius. Test runs with one half of that time step and one quarter confirmed the choice. I used random number generators taken from Press et al. (1992).

[Reference]

REFERENCES

[Reference]

Berridge, M., M. Bootman, and P. Lipp. 1998. Calcium - a life and death signal. Nature. 395:645-648.

Bootman, M., M. Berridge, and P. Lipp. 1997a. Cooking with calcium: The recipes for composing global signals from elementary events. Cell. 91:367-373.

Bootman, M., E. Niggli, M. Berridge, and P. Lipp. 1997b. Imaging the hierarchical Caz+ signaling in HeLa cells. J. Physiol. 499:307-314.

Callamaras, N., and I. Parker. 2000. Phasic characteristics of elementary Ca2+ release sites underlies quantal responses to IP3. EMBO J. 19-160P-1617

[Reference]

Callamaras, N., J. Marchant, X.-P. Sun, and I. Parker. 1998. Activation and coordination of insp3-mediated elementary Ca2+ events during global Caz+ signals in Xenopus oocytes. J. Physiol. (Lond.). 509:81-91.

DeYoung, G., and J. Keizer. 1992. A single-pool inositol 1,4, 5-trisphosphate-receptor-based model for agonist-stimulated oscillations in Ca2+ concentration. Proc. Natl. Acad. Sci. USA. 89: 9895-9899.

[Reference]

Dupont, G., and S. Swillens. 1995. Quantal release, incremental detection, and long-period Ca2+ oscillations in a model based on regulatory Ca2+ -binding sites along the permeation pathway. Biophys. J. 71: 1714-1722.

Falcke, M. 2003. Buffers and oscillations in intracellular Caz+ dynamics. Biophys. J. 84:28-41.

Falcke, M., J. Hudson, P. Camacho, and J. Lechleiter. 1999. Impact of mitochondrial Caz+ cycling on pattern formation and stability. Biophys. J. 77:37-44.

[Reference]

Falcke, M., M. Or-Guil, and M. Bar. 2000a. Dispersion gap and localized spiral waves in a model for intracellular Ca + dynamics. Phys. Rev. Left. 84:4753-4756.

Falcke, M., L. Tsimring, and H. Levine. 2000b. Stochastic spreading of intracellular Ca2+ release. Phys. Rev.E. 62:2636-2643.

Gall, D., E. Baus, and G. Dupont. 2000. Activation of the liver glycogen phosphorylase by Ca2+ oscillations: a theoretical study. J. Theor. Biol. 207:445-454.

[Reference]

Garca-Ojalvo, J., and L. Schimansky-Geier. 2000. Excitable structures in stochastic bistable media. J. Stat. Phys. 101:473-481.

Hempel, H., L. Schimansky-Geier, and J. Garcia-Ojalvo. 1999. Noise-- sustained pulsating patterns and global oscillations in subexcitable media. Phys. Rev. Lett. 82:3713-3716.

Ilyin, V., and I. Parker. 1994. Role of cytosolic Caz+ in inhibition of InsP3evoked Ca2+ release in Xenopus oocytes. J. Physiol. 477:503-509. Jung, P., and P. Galley. 2000. The heartbeat of extended clocks. Ann. Phys. 9:697-704.

[Reference]

Keizer, J., and G. DeYoung. 1994. Simplification of a realistic model of IP3-induced Ca2+ oscillations. J. Theor. Biol. 166:431-442.

LeBeau, A., D. Yule, G. Groblewski, and J. Sneyd. 1999. Agonist-- dependent phosphorylation of the inositol 1,4,5-trisphosphate receptor a possible mechanism for agonist-specific calcium oscillations in pancreatic acinar cells. J. Gen. Physiol. 113:851-871.

Lechleiter, J., S. Girard, E. Peralta, and D. Clapham. 1991. Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes. Science. 252:123-126.

Lukyanenko, V., and S. Gy/rke. 1999. Ca2+ sparks and Ca2+ waves in saponin-permeabilized rat ventricular myocytes. J. Physiol. 521:575-585.

[Reference]

Mak, D., and J. Foskett. 1998. Effects of divalent cations on single-channel conduction properties of Xenopus IP3 receptor. Am. J. Physiol. 275:C179-C188.

Mak, D., S. McBride, and J. Foskett. 2001. Regulation by Caz+ and Inositol 1,4,5-trisphosphate (InsP3 of single recombinant type 3 InsP3 receptor channels: Activation uniquely distinguishes type 1 and type 3 InsP3 receptors. J. Gen. Physiol. 117:435-446.

Mak, D., S. McBride, V. Raghuram, Y. Yue, S. Joseph, and J. Foskett. 2000. Single-channel properties in endoplasmic reticulum membrane of recombinant type 3 inositol trisphosphate receptor. J. Gen. Physiol. 115:241-255.

[Reference]

Marchant, J., and I. Parker. 2001. Role of elementary Ca2+ puffs in generating repetitive Caz+ oscillations. EMBO J. 20:65-76.

Marchant, J., N. Callamaras, and I. Parker. 1999. Initiation of IP3-mediated Ca2+ waves in Xenopus oocytes. EMBO J. 18:5285-5299.

Moraru, I., E. Kaftan, B. Ehrlich, and J. Watras. 1999. Regulation of type 1 Inositol 1,4,5-trisphosphate-gated calcium channels by InsP3 and calcium. J. Gen. Physiol. 113:837-849.

Olveczky, B., and A. Verkman. 1998. Monte carlo analysis of obstructed diffusion in three dimensions: application to molecular diffusion in organelles. Biophys. J. 74:2722-2730.

Patel, S., S. Joseph, and A. Thomas. 1999. Molecular properties of inositol 1,4,5-trisphosphate receptors. Cell Calcium. 25:247-264.

Press, W., S. Teukolsky, W. Vetterling, and B. Flannery. 1992. Numerical recipes in C. 2nd ed, Cambridge University Press, Cambridge.

Shuai, J., and P. Jung. 2002. Optimal intracellular calcium signaling. Phys. Rev. Lett. 88:U252-U255.

[Reference]

Sneyd, J., and J.-F. Dufour. 2002. A dynamic model of the type-2 inositol trisphosphate receptor. Proc. Natl. Acad. Sci. USA. 99:2398-2403.

Sneyd, J., J. Keizer, and M. Sanderson. 1995. Mechanisms of calcium oscillations and waves: a quantitative analysis. FASEB J. 9:1463-1472.

Sun, X.-P., N. Callamaras, J. Marchant, and 1. Parker. 1998. A continuum of insp3-mediated elementary CaZ+ signaling events in Xenopus oocytes. J. Physiol. (Lond.). 509:67-80.

Swillens, S., P. Champed, L. Combettes, and G. Dupont. 1998. Stochastic simulation of a single inositol 1,4,5-trisphoasphate-sensitive Caz+ channel reveals repetitive openings during blip-like Caz+ transients. Cell Calcium. 23:291-302.

[Reference]

Tang, Y., and H. Othmer. 1996. Simplification and analysis of models of calcium dynamics based on IP3-sensitive calcium channel kinetics. Biophys. J. 70:246-263.

Taylor, C. 1998. Inositol trisphosphate receptors: Ca2+-modulatd intracellular Caz+ channels. Biochim. Biophys. Acta. 1436:19-33.

Thomas, D., P. Lipp, M. Berridge, and M. Boorman. 1998. Hormonestimulated elementary Ca2+ signals are not stereotypic, but reflect activation of different size channel clusters and variable recruitment of channels within a cluster. J. Biol. Chem. 273:27130-27136.

Thomas, D., P. Lipp, S. Tovey, M. Berridge, W. Li, R. Tsien, and M. Boorman. 1999. Microscopic properties of elementary Ca2+ release sites in non-excitable cells. Curr. Biol. 10:8-15.

Tsien, R., and R. Tsien. 1990. Calcium channels stores and oscillations. Annu. Rev. Cell Biol. 6:715-760.

Wagner, J., Y.-X. Li, J. Pearson, and J. Keizer. 1998. Simulation of the fertilization Caz+ wave in Xenopus laevis eggs. Biophys. J. 75:2088-2097.

[Author Affiliation]

Martin Falcke

Hahn Meitner Institute, Glienicker Str. 100, 14109 Berlin, Germany and Max Planck Institute for the Physics of Complex Systems, Nothnitzer Str. 38, 01187 Dresden, Germany

[Author Affiliation]

Submitted June 25, 2002, and accepted for publication August 30, 2002.

Address reprint requests to M. Falcke, Hahn Meitner Institute, Glienicker Str. 100, 14109 Berlin, Germany. Tel.: 49-30-80622627; Fax: +49-3080622098; E-mail: falcke@hmi.de.