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									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
Ca2+ oscillations in various types of
        nonexcitable cells




  Hepatocytes        Oocytes




  Astrocytes         Pancreatic acinar cells
                                                         Effect 1

                                                         Effect 2
 Vasopressin
                                            Calmodulin   Effect 3
                     Ca2+   oscillation
Phenylephrine                                Calpain


   Caffeine                                    PKC


       UTP                                    …..




          Bow-tie structure of signalling

          How can one signal transmit several signals?
     Scheme of main processes
                  H
                                                                 Caext
                      R          PLC                              vin
                                 vplc vd
                               DAG IP3
                          PIP2
        vout                                      cytosol
                                       +
                          vrel                             vmi           vmo

         vserca                        +
                                              Cacyt                     Cam
                                 ER                              mitochondria
                          Caer                      vb,j


                                       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:
  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
                                                   H
                                                                                                 Caext
                                                       R          PLC                             v1
                                                                    vplc     vd
                                                           PIP2
                                        v2                        DAG IP3          cytosol
                                                                        +
Influx into the cell:    v1  const .                      v5                              vmi          vmo
                                                                            Cacyt=S1
                                             v4                         +                              Cam
                                                                  ER                             mitochondria
                                                       Caer=S2
Efflux out of the cell: v2  k2 S1                                                  vb,j


                                                  v6
                                                                        proteins
Pumping of    Ca2+   into ER: v4  k 4 S1

                                                                      4
                                                                k5S2 S1
Efflux out of ER through channels (CICR):         v5              4
                                                            K 4  S1

Leak out of the ER:      v6  k6S2
       Temporal behaviour




                    fast movement   Relaxation
                                    oscillations!



slow movement
                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




                                            variable
variable




                                                              unstable limit cycle
                       stable limit cycle                                  stable limit cycle



                  parameter                                         parameter

                                                           Hysteresis!
             Bifurcation diagram
           for calcium oscillations




                                           From: S. Schuster &
                  oscillations             M. Marhl, J. Biol. Syst.
                                           9 (2001) 291-314


Subcritical HB                   Supercritical HB
Schematic picture of bifurcation diagram



             variable


                                  parameter

                        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
        Before bifurcation
S2
                                             At bifurcation




                             Saddle point                           Saddle point


       Unstable focus
                                                               Homoclinic orbit
                                    S1
After bifurcation
                                             Necessary condition in 2D systems:
                                             at least 2 steady states
                                             (in Somogyi-Stucki model,
                                             only one steady state)
                              Saddle point


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

                  H
                                                          Caext
                   R          PLC                          vin
                                vplc    vd
                       PIP2
         vout                 DAG IP3          cytosol
                                    +
                       vrel                         vmi           vmo

         vserca                     +                            Cam
                                             Cacyt
                              ER                          mitochondria
                       Caer                      vb,j


                                    proteins




          Marhl, Schuster, Brumen, Heinrich, Biophys. Chem. 63 (1997) 221
System equations
dCacyt
          J ER, ch  J ER, pump  J ER, leak  J CaPr  J Pr
  dt

dCaER  ER
           ( J ER, pump  J ER, ch  J ER, leak )                 2D model
  dt    ER


 with
                     2
           ~       Cacyt
J ER, ch  g Ca                  ( ECa   )
                   2
                  K1    Cacyt
                           2


Nonlinear equation for transmembrane potential ...

J ER, pump  k ER, pumpCacyt           J ER, leak  k ER, leak (CaER  Cacyt )

J Pr  k  Cacyt Pr          J CaPr  k CaPr
…this gives rise to a
homoclinic bifurcation

                               oscillation
              Hopf bifn.
   variable




                              parameter      Schuster &
               Saddle point
                                             Marhl,
                                             J. Biol. Syst.
 As the velocity of the trajectory           9 (2001) 291
 tends to zero when it approaches
 the saddle point, the oscillation
 period becomes arbirtrarily long
 near the bifurcation.
    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




           Prot1T * Ca 4                     Prot2T * Ca 4
Prot1Ca4                  Prot2Ca4 
            K1  Ca 4                                 Ca 4 
                                      ( K 2  Ca ) * 1 
                                                4
                                                              
                                                          KI 
Selective activation of protein 1


                        Prot1




                         Prot2
Selective activation of protein 2



                       Prot2




                       Prot1
Simultaneous up- and
    downregulation


  Prot2   Prot1




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




                                      kon = 1 s-1mM-4




       koff/kon = const. = 0.01 mM4
               „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.

								
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