Stefan Schuster and Beate Knoke
Dept. of Bioinformatics
Friedrich Schiller University Jena
Germany

• 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
Hepatocytes
Astrocytes
Oocytes
Pancreatic acinar cells

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

v out v serca
H
R
PIP
2
PLC
DAG v plc IP
3 v d
+ v rel cytosol v mi
Ca ext v in v mo
Ca er
ER
+ Ca cyt v b,j
Ca m mitochondria proteins
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
H
Influx into the cell:
Efflux out of the cell: v
1
 const .
v
2
 k S v
2 v
4
Ca ext
R
Ca
PIP
2 v
5 er
PLC v plc
DAG
IP
3 v d cytosol
+ v mi
+
Ca cyt
=S
1
=S
ER
2 v b,j v
1 v mo
Ca m mitochondria v
6
Pumping of Ca 2+ into ER: v
4
 k
4
S
1 proteins
Efflux out of ER through channels (CICR): v
5
 k S S
4
5 2 1
K
4 
S
1
4
Leak out of the ER: v
6
 k S

Temporal behaviour slow movement fast movement
Relaxation oscillations!

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, 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 stable limit cycle parameter unstable limit cycle stable limit cycle parameter
Hysteresis!

Bifurcation diagram for calcium oscillations
Subcritical HB oscillations
From : S. Schuster &
M. Marhl, J. Biol. Syst.
9 (2001) 291-314
Supercritical HB

Schematic picture of bifurcation diagram
Bifurcation parameter
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

S
2
Homoclinic bifurcation
Before bifurcation
At bifurcation
Unstable focus
After bifurcation
Saddle point
Saddle point
Homoclinic orbit
S
1
Saddle point
Necessary condition in 2D systems: at least 2 steady states
(in Somogyi-Stucki model, only one steady state)
Limit cycle

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

System equations d Ca cyt 
J
ER, ch

J
ER, pump

J
ER, leak

J
CaPr

J
Pr d t d Ca
ER d t



ER
ER
( J
ER, pump

J
ER, ch

J
ER, leak
)
2D model with
J
ER, ch
 g
Ca
K
1
2
Ca
2 cyt

Ca
2 cyt
( E
Ca
 
)
Nonlinear equation for transmembrane potential
...
J
ER, pump
 k
ER, pump
Ca cyt
J
ER, leak
 k
ER, leak
( Ca
ER

Ca cyt
)
J
Pr
 k  Ca cyt
Pr J
CaPr
 k

Ca
Pr

…this gives rise to a homoclinic bifurcation oscillation
Hopf bifn.
Saddle point parameter
As the velocity of the trajectory tends to zero when it approaches the saddle point, the oscillation period becomes arbirtrarily long near the bifurcation.
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
Prot Ca
1 4

Prot
1T
* Ca
4
K
1

Ca
4
Prot Ca
2 4

( K
2

Prot Ca
2T
*
Ca
4

) * 1

4
Ca
4
K
I



Selective activation of protein 1
Prot
1
Prot
2

Selective activation of protein 2
Prot
2
Prot
1

Simultaneous up- and downregulation
Prot
2 Prot
1
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 k on k on
= 500 s -1 mM -4
= 15 s -1 mM -4 k on
= 1 s -1 m
M -4 k off
/ k on
= const. = 0.01 m
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

• 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.

• 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

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