Modeling of Calcium Signaling PathwaysStefan Schuster and Beate Knoke

Dept. of BioinformaticsFriedrich 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

Ca2+ 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?

Ca2+ oscillation

Scheme of main processes

PLCR

H

vout

vin

cytosol+

vrel

vserc

a

IP3

mitochondria

vmivmo

CamCacyt

vb,j

proteins

vplcvd

Caext

+

PIP2 DAG

ERCaer

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: Ca2+ in cytosol (S1) and Ca2+ in endoplasmic reticulum (S2)

• 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

v const1 .

v k S2 2 1

144 Skv

Influx into the cell:

Efflux out of the cell:

Pumping of Ca2+ into ER:

Efflux out of ER through channels (CICR): vk S S

K S5

5 2 14

414

Leak out of the ER: v k S6 6 2

PLCR

H

v2

v1

cytosol+

v5

v4

IP3

mitochondria

vmivmo

Cam

Cacyt=S1

vb,j

proteins

vplcvd

Caext

+

PIP2 DAG

ERCaer=S2

v6

fast movement

slow movement

Relaxationoscillations!

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, 1333-1355 and Falcke, M.

Adv. Phys. (2004) 53, 255-440.

• 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, k5 (CICR),

dependent on IP3

• Low k5 : steady state; medium k5: oscillations; high k5:

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

varia

ble

parameterva

riabl

e

stable limit cycle 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) 291-314

oscillations

Schematic picture of bifurcation diagram

parameter

varia

ble

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 Saddle point

Homoclinic orbit

Before bifurcationAt 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

S1

S2

Model including binding of Ca2+ to proteins and effect of ER transmembrane potential

PLCR

H

vout

vin

cytosol+

vrel

vserca

IP3

mitochondria

vmivmo

CamCacyt

vb,j

proteins

vplcvd

Caext

+

PIP2 DAG

ERCaer

Marhl, Schuster, Brumen, Heinrich, Biophys. Chem. 63 (1997) 221

PrCaPrleak ER,pump ER,ch ER,cyt

d

dJJJJJ

t

Ca

)(d

dleak ER,ch ER,pump ER,

ER

ERER JJJt

Ca

System equations

with

)(~2cyt

21

2cyt

ch ER,

CaCa ECaK

CagJ

cytpump ER,pump ER, CakJ )( cytERleak ER,leak ER, CaCakJ

PrcytPr CakJ PrCakJ CaPr

Nonlinear equation for transmembrane potential

2D model

…this gives rise to ahomoclinic bifurcation

varia

ble

Saddle point

oscillation

As the velocity of the trajectory tends to zero when it approachesthe saddle point, the oscillationperiod becomes arbirtrarily longnear 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 Ca2+ - binding proteins

41T

1 4 41

*Prot CaProt Ca

K Ca

42T

2 4 44

2I

*

( )* 1

Prot CaProt Ca

CaK Ca

K

Selective activation of protein 1

Prot1

Prot2

Selective activation of protein 2

Prot1

Prot2

Simultaneous up- and downregulation

Prot1Prot2

S. Schuster, B. Knoke, M. Marhl: Differential regulation of proteins by bursting calcium oscillations – A theoretical study. BioSystems 81 (2005)49-63.

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

kon = 1 s-1M-4

kon = 15 s-1mM-4kon = 500 s-1mM-4

koff/kon = const. = 0.01 M4

„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.