Modeling of Calcium Signaling Pathways Stefan Schuster and Beate Knoke Dept. of Bioinformatics Friedrich Schiller University Jena Germany

1. Introduction Oscillations of intracellular calcium ions are important in signal transduction both in excitable and nonexcitable cells A change in agonist (hormone) level can lead to a switch between oscillatory regimes and stationary states  digital signal Moreover, analogue signal encoded in frequency Amplitude encoding and the importance of the exact time pattern have been discussed; frequency encoding is main paradigm

Ca 2+ oscillations in various types of nonexcitable cells Astrocytes Hepatocytes Oocytes Pancreatic acinar cells

Vasopressin Phenylephrine Caffeine UTP Calmodulin Calpain PKC ….. Effect 1 Effect 2 Effect 3 Bow-tie structure of signalling How can one signal transmit several signals? Ca 2+ oscillation

Scheme of main processes PLC R H v out v in cytosol + v rel v serca IP 3 mitochondria v mi v mo Ca m Ca cyt v b,j proteins v plc vdvd Ca ext + PIP 2 DAG ER Ca er Efflux of calcium out of the endoplasmic reticulum is activated by cytosolic calcium = calcium induced calcium release = CICR

Somogyi-Stucki model Is a minimalist model with only 2 independent variables: Ca 2+ in cytosol (S 1 ) and Ca 2+ in endoplasmic reticulum (S 2 ) All rate laws are linear except CICR R. Somogyi and J.W. Stucki, J. Biol. Chem. 266 (1991) 11068

Rate laws of Somogyi-Stucki model Influx into the cell: Efflux out of the cell: Pumping of Ca 2+ into ER: Efflux out of ER through channels (CICR): Leak out of the ER: PLC R H v2v2 v1v1 cytosol + v5v5 v4v4 IP 3 mitochondria v mi v mo Ca m Ca cyt =S 1 v b,j proteins v plc vdvd Ca ext + PIP 2 DAG ER Ca er =S 2 v6v6

fast movement slow movement Relaxation oscillations! Temporal behaviour

Many other models… by A. Goldbeter, G. Dupont, J. Keizer, Y.X. Li, T. Chay etc. Reviewed, e.g., in Schuster, S., M. Marhl and T. Höfer. Eur. J. Biochem. (2002) 269, and Falcke, M. Adv. Phys. (2004) 53, Most models are based on calcium-induced calcium release.

2. Bifurcation analysis of two models of calcium oscillations Biologically relevant bifurcation parameter in Somogyi- Stucki model: rate constant of channel, k 5 (CICR), dependent on IP 3 Low k 5 : steady state; medium k 5 : oscillations; high k 5 : steady state. Transition points (bifurcations) between these regimes can here be calculated analytically, be equating the trace of the Jacobian matrix with zero.

Usual picture of Hopf bifurcations Supercritical Hopf bifurcation Subcritical Hopf bifurcation parameter variable parameter variable stable limit cycle unstable limit cycle Hysteresis!

Bifurcation diagram for calcium oscillations Subcritical HB Supercritical HB From: S. Schuster & M. Marhl, J. Biol. Syst. 9 (2001) oscillations

Schematic picture of bifurcation diagram parameter variable Bifurcation Very steep increase in amplitude. This is likely to be physiologically advantageous because oscillations start with a distinct amplitude and, thus, misinterpretation of the oscillatory signal is avoided. No hysteresis – signal is unique function of agonist level.

Global bifurcations Local bifurcations occur when the behaviour near a steady state changes qualitatively Global bifurcations occur „out of the blue“, by a global change Prominent example: homoclinic bifurcation

Homoclinic bifurcation Saddle point Homoclinic orbit Before bifurcation At bifurcation After bifurcation Limit cycle Saddle point Necessary condition in 2D systems: at least 2 steady states (in Somogyi-Stucki model, only one steady state) Unstable focus S1S1 S2S2

Model including binding of Ca 2+ to proteins and effect of ER transmembrane potential PLC R H v out v in cytosol + v rel v serca IP 3 mitochondria v mi v mo Ca m Ca cyt v b,j proteins v plc vdvd Ca ext + PIP 2 DAG ER Ca er Marhl, Schuster, Brumen, Heinrich, Biophys. Chem. 63 (1997) 221

System equations with Nonlinear equation for transmembrane potential  2D model

…this gives rise to a homoclinic bifurcation variable Saddle point oscillation As the velocity of the trajectory tends to zero when it approaches the saddle point, the oscillation period becomes arbirtrarily long near the bifurcation. parameter Hopf bifn. Schuster & Marhl, J. Biol. Syst. 9 (2001) 291

3. How can one second messenger transmit more than one signal? One possibility: Bursting oscillations (work with Beate Knoke and Marko Marhl)

Differential activation of two Ca 2+ - binding proteins

Selective activation of protein 1 Prot 1 Prot 2

Selective activation of protein 2 Prot 1 Prot 2

Simultaneous up- and downregulation Prot 1 Prot 2 S. Schuster, B. Knoke, M. Marhl: Differential regulation of proteins by bursting calcium oscillations – A theoretical study. BioSystems 81 (2005)

4. Finite calcium oscillations Of course, in living cells, only a finite number of spikes occur Question: Is finiteness relevant for protein activation (decoding of calcium oscillations)?

Intermediate velocity of binding is best k on = 1 s -1  M -4 k on = 15 s -1 mM -4 k on = 500 s -1 mM -4 k off /k on = const. = 0.01  M 4

„Finiteness resonance“ Proteins with different binding properties can be activated selectively. This effect does not occur for infinitely long oscillations. M. Marhl, M. Perc, S. Schuster S. A minimal model for decoding of time-limited Ca(2+) oscillations. Biophys Chem. (2005) Dec 7, Epub ahead of print

5. Discussion Relatively simple models (e.g. Somogyi-Stucki) can give rise to complex bifurcation behaviour. Relaxation oscillators allow jump-like increase in amplitude at bifurcations and do not show hysteresis. At global bifurcations, oscillations start with a finite (often large) amplitude. Physiologically advantageous because misinterpretation of the oscillatory signal is avoided in the presence of fluctuations.

Discussion (2) Near homoclinic bifurcations, oscillation period can get arbitrarily high. This may be relevant for frequency encoding. Frequency can be varied over a wide range. Bursting oscillations may be relevant for transmitting two signals simultaneously – experimental proof is desirable Thus, complex oscillations as found in, e.g. hepatocytes, may be of physiological importance Finite trains of calcium spikes show resonance in protein activation Thus, selective activation of proteins is enabled

Cooperations Marko Marhl (University of Maribor, Slovenia) Thomas Höfer (Humboldt University, Berlin, Germany) Exchange with Slovenia supported by Research Ministries of both countries.