May 27, 2005 Control of stability of intracellular Ca-oscillations and electrical activity in a network of coupled cells. Stan Gielen Dept. of Biophysics

May 27, 2005

Control of stability of intracellular Ca-oscillations and electrical

activity in a network of coupled cells.

Stan GielenDept. of Biophysics

Martijn Kusters

Wilbert van Meerwijk

Dick Ypey

Lex Theuvenet

May 27, 2005

Overview• Summary of Ca-dynamics in NRK cell• Dynamics of Ca-oscillations and action potentials• coupling between Ca-oscillations and action

potentials• Stability of Ca-dynamics in the cell• Alternative model for cells with IP3-oscillations• Coupling between two oscillators• Propagation of electrical activity in network of

layers– oscillators as pacemakers which initiate propagation ?– instability due to coupling ?

May 27, 2005

Model for Normal Rat Kidney Cell

• NRK-cell = fibroblast

• similar to Cells of Cajal

• NRK cells form a network coupled by gap-junctions

May 27, 2005

Model for Membrane NRK cell

May 27, 2005

Components of the model

CacytBCacyt

B

Caex

GCaLClex

GCl(Ca)

KcytGKir

Gleak

ATPPMCA

CaER

(0.1 μM)

(1000 μM)

(1000 μM)

May 27, 2005

This model focusses on the dynamics of the cell membrane, including the L-type Ca-channel and other ion channels with the following components:

• PMCA pump : pump Ca out of cytosol into extracellular space

• Ca2+ L-type channel: Vca-L = +55 mV• Cl(Ca) channel : VCl = -20 mV• Leak channel• Kir channel : VK = -75 mV• Ca-buffer in the cytosol Cacyt BCacyt

B

Caex

GCaLClex

GCl(Ca)

KcytGKir

Gleak

PMCA

CaER

May 27, 2005

Components of the model for the NRK Membrane

• Leak current

• Potassium channel

• PMCA-pomp

PMCAcyt

cytPMCAPMCA

OK

K

KKK

KK

KKK

KOKK

leakleakleak

KCa

CaCJ

K

F

RTE

EV

EVEV

EV

EVK

GI

EVGI

)120

ln(1000

))50(06.0exp(1

)10(0002.0exp))100(0002.0exp(3

})50{06.0exp(1

1.0

)(4.5

)(

May 27, 2005


• Ca2+ L type channel

• Cl(Ca) kanaal)(

1

1

))}10(0337.0{exp(0197.002.0

01.0

)24.5/)37exp((1

1

)10(035.0

))9.5/)10(exp(1(01.0

)24.5/)15(exp(1

1

)(

)()()(

2

)()()(

CaClClcyt

cytCaClCaCl

cytwCaCa

h

m

LCaCaLCaLCa

EVKCa

CaGI

CaKw

V

Vh

V

Vm

Vm

EVwhmGI

May 27, 2005

Current clamp Ipulse=6 pA• When we current clamp, the activation gate of the Ca L

type opens, giving rise to an inflow of Ca through the Ca L type channel.

• As a consequence, an action potential will be generated

Cacyt

BCacyt

B

Caex

GCaLGCl(Ca)

K

GKir

Gleak

PMCA

CaER

May 27, 2005

Current clamp Ipulse=6 pAAction potentialCacyt

Buffered Ca

PMCA current

IK

ICl

ILeak

)20(

)55(

)()(

)()(

mVVKCa

CaGI

mVVwhmGI

Clcyt

cytCaClCaCl

CaLCaLCa

PMCAcyt

cytPMCAPMCA

KKK

KOKK

leakleakleak

KCa

CaCJ

EVK

GI

EVGI

)(4.5

)(

May 27, 2005


Buffered Ca

PMCA current

IK

ICl

ILeak

Inflow of Ca through L-type Ca channel

Plateau due to Nernst potential of Ca-dependent Cl-channel

May 27, 2005


Buffered Ca

PMCA current

IK

ICl

ILeak

)20(

)55(

)()(

)()(

mVVKCa

CaGI

mVVwhmGI

Clcyt

cytCaClCaCl

CaLCaLCa

PMCAcyt

cytPMCAPMCA

KKK

KOKK

leakleakleak

KCa

CaCJ

EVK

GI

EVGI

)(4.5

)(

Important !

May 27, 2005

Adding a Ca2+ buffer eliminates the plateau

De Roos et al. 1998

May 27, 2005

The effect of a Ca-bufferWith Ca buffer Without Ca buffer

Shorter plateau-phase

May 27, 2005

Model for intracellular Ca2+-oscillations

May 27, 2005

Model for Ca-oscillations from ER

Glek

CaER

Cacyt

BCacyt

BGlk

Glek

ATP PMCA

IP3 receptor

ATP

ATP

SERCA

• SERCA pump• IP3-receptor• leakage of Ca from the ER into the cytosol• PMCA pump• leakage of Ca from extracellular space into the cytosol• Ca-buffer in the cytosol

(0.1 μM)

(1000 μM)

May 27, 2005

This model focusses on the dynamics of Ca in the ER and cytosol by transport through the IP3 receptor. The model has the following components:

• SERCA pump• IP3-receptor• leakage of Ca from the ER into the cytosol• PMCA pump• leakage of Ca from extracellular space into the

cytosol• Ca-buffer in the cytosol

May 27, 2005

Components of the model for the IP3-oscillator

• IP3-receptor

• Leakage from ER• SERCA-pomp

cytwIP

w

w

cytwIP

w

wIPw

wIPfIPcyt

cyt

CaKIP

IPK

CaKIP

IPK

KIPIP

K

w

KIP

IP

KCa

Caf

1.03

320

333

3

3

3

3

32

32

313

)(,, cytERERleakERleak CaCaCJ

)(3333 cytERIPIP CaCawfCJ

A conversion factor of 0.1 transforms an increase/decrease of CaER into a decrease/increase of Cacyt.

May 27, 2005

Intracellular Ca-oscillations

Harks et al., 2004

May 27, 2005

Stability analysis of IP3 receptor

cytwIP

w

w

cytwIP

w

wIPw

wIPfIPcyt

cyt

CaKIP

IPK

CaKIP

IPK

KIP

IPK

w

KIP

IP

KCa

Caf

1.03

320

3

33

3

3

3

3

32

32

313

May 27, 2005

Ca-oscillations as a function of IP3

May 27, 2005

IP3-mediated calcium oscillations

Ca E

RBuf

fere

d C

aBuffered Ca

CacytC

a cyt

CaER

JPMCA,JSOC,JLeakJSERCA, JIP3

May 27, 2005

Concentration IP3 low high

IP3-mediated calcium oscillations

CaERCaERCacyt CacytBuffered Ca Buffered Ca

JSERCA, JIP3JSERCA, JIP3JPMCA,JSOC,JLeak JPMCA,JSOC,JLeak

May 27, 2005




May 27, 2005

Stability of Ca-dynamics in the cell

Whole cell modelAction potentials

Ca-oscillations

May 27, 2005

CacytBCacyt

B

IP3

Caex

GCaLClex

GCl(Ca)

KcytGKir

Glk

GCalk PMCA

CaerSERCA

IP3RJCalker

Complete Model

May 27, 2005

steady-state behavior

Without IP3, the steady-state is easily found by solving JSERCA=Jleak,ER and JPMCA=Jleak,membrane

This gives a single, stable solution for Cacyt and CaER :

Cacytosol = 0.1 μM; CaER= 1300 μM

)1000(maxcytlk

PMCAcyt

cytPMCA CaG

KCa

CaI

SERCAcyt

cytSERCAcytERleak KCa

CaJCaCaG

max)(

ER/cytosol:

membrane/cytosol:

May 27, 2005

Stability of Ca2+ concentrations

Cacyt

Caex (1000 μM)

GCalk PMCA

CaerSERCAJCalker

Cacyt

GCalk PMCA

CaerSERCAJCalker

Action potential triggers Ca oscillation Ca oscillation triggers action potential

Caex (1000 μM)

May 27, 2005

CacytBCacyt

B

IP3

Caex

GCaLClex

GCl(Ca)

KcytGKir

Glk

GCalk PMCA

CaerSERCA

IP3RJCalker

Additional channel to stabilize Ca-dynamics

GSOC

May 27, 2005

Whole cell model with SOC/CRAC channel

Action potentials

Ca-oscillations

May 27, 2005


Mem

bran

e po

tent

ial

Cacytosol (μMol)

dV/dt = 0

dCacyt/dt=0

IP3 = 0

Stable attractor

May 27, 2005


Mem

bran

e po

tent

ial

dV/dt = 0

dCacyt/dt=0

IP3 receptor oscillates

No stable attractor

0 0.5 1 1.5 2

Red for ca

80

60

40

20

0

20

40

60

eulB

rofV

ip3 1

Cacytosol (μM)

May 27, 2005


dV/dt = 0

dCacyt/dt=0

IP3 high

Stable attractor at – 20 mV

0 0.5 1 1.5 2

Red for ca

80

60

40

20

0

20

40

60

eulB

rofV

ip3 3

Mem

bran

e po

tent

ial

Cacytosol (μMol)

May 27, 2005

Stability analysis of IP3 receptor

cytwIP

w

w

cytwIP

w

wIPw

wIPfIPcyt

cyt

CaKIP

IPK

CaKIP

IPK

KIP

IPK

w

KIP

IP

KCa

Caf

1.03

320

3

33

3

3

3

3

32

32

313

May 27, 2005

Summary

• Stability of Ca-dynamics for all possible natural conditions requires a coupling between Ca-concentration in ER and extracellular Ca.

• Without IP3: stable condition corresponds to V=-70 mV; Cacyt=0.1 μM

• Higher IP3 concentrations provide oscillations or stable point at V= -20 mV

May 27, 2005




May 27, 2005

Alternative model for coupling between IP3-oscillator

(Ca-oscillations) and membrane oscillator (action potentials)

May 27, 2005

ProblemMany cell types do not oscillate in isolation, but do so in a

synchronized manner only when electrically coupled in a network (e.g. β-pancreatic cells in islets of Langerhans and aortic smooth muscle cells).– Cells in isolation are quiet or oscillate at lower

frequencies.Paradox: If identical cells oscillate in phase, there are no

currents ! How then can electrical coupling be crucial for the synchronous oscillations ? Moreover: if there are phase differences, they will be eliminated by the electrical coupling !

May 27, 2005

Basic mechanism

0),(

),(

),(

cyt

ERcyt

ERcytER

cytERcytcyt

dCa

CaCadJ

CaCaJdt

dCa

UKCaCaCaJdt

dCa

J(Cacyt,CaER) = interaction term between Ca concentrations

with reflecting Ca-induced Ca-release

KCacyt = efflux of Ca from cell

U = constant, Ca-mediated electrical current

0),(

cyt

ERcyt

dCa

CaCadJ

Loewenstein & Sompolinsky, PNAS, 2001

May 27, 2005

Calcium and Voltage oscillationsin non-excitable cell

Cytosolic Ca (μM) Ca in stores (μM)

Rest-state is unstable fixed-point

Small perturbations in cytosolic Ca cause oscillations

Loewenstein et al., PNAS 98, 2001

May 27, 2005

Calcium and Voltage oscillationsCytosolic Ca (μM) Ca in stores (μM)

Non-excitable cell

Excitable cell

with Voltage-dependent Ca-current en Kca channel

IK_Ca hyperpolarizes membrane potential, which de-activates Ca-influx into cell

However, adding a shunt conductance

destabilizes the fixed point

Hyperpolarization decreases by electrical coupling )( ji

jij

icoupling VVgI

May 27, 2005

Calcium and Voltage oscillationsCytosolic Ca (μM) Ca in stores (μM)

Excitable cell

with Voltage-dependent Ca-current en Kca channel

Addition of ashunt conductance

1. Reduces the effect of Cacyt on membranbe potential

2. Suppresses efficacy of negative feedback by IK_Ca

3. Enables oscillations

with Voltage-dependent Ca-current en Kca channel but with shunt conductance

May 27, 2005

Voltage and Ca oscillations in network of two electrically coupled cells

Ca oscillations out-of-phase; electrical oscillations in-phase at double frequency

Hyperpolarization due to Ca-influx

Hyperpolarization due to electrical coupling

May 27, 2005

Multi-stability in network with 6 coupled

cells.

Cell

123456

123456

In a large network different realizations of out-of-phase calcium oscillations are possible and therefore the network possesses many stable states. The stable state in which the system will eventually settle is determined by the initial conditions.

Note the differences in membrane potential !

May 27, 2005

Summary• Cells are

– intrinsically stable (near –70 mV ; Loewenstein et al. PNAS 2001) or

– intrinsically oscillating ?

• Electrical coupling – enables oscillations and propagation of activity to

otherwise silent cells or – disables oscillations and propagating activity in a

network of pacemaker cells ?

• Ca oscillations out of phase ! Why ?

May 27, 2005




May 27, 2005

Coupling between two oscillators

Inhibition and electrical coupling

May 27, 2005

Neuronal synchronization due to external inputT

ΔTΔ(θ)= ΔT/T

Synaptic input

May 27, 2005

Neuronal synchronizationT

ΔTΔ(θ)= ΔT/T

Phase shift as a function of the relative phase of the external input.

Phase advance

Hyperpolarizing stimulus

Depolarizing stimulus

May 27, 2005

Neuronal synchronizationT

ΔTΔ(θ)= ΔT/T

Suppose:

• T = 95 ms

• external trigger: every 76 ms

• Synchronization when ΔT/T=(95-76)/95=0.2

• external trigger at time 0.7x95 ms = 66.5 ms

May 27, 2005

Inhibitory couplingfor two identical leaky-integrate-and-fire neurons

Out-of-phase stable In-phase stable

Lewis&Rinzel, J. Comp. Neurosci, 2003

May 27, 2005

Phase-shift functionfor inhibitory coupling

0)( *

d

dG

for stable attractor

Increasing constant input to the LIF-neurons

I=1.2

I=1.4

I=1.6

May 27, 2005

Bifurcation diagram for two identical LIF-neurons with inhibitory coupling

May 27, 2005

Bifurcation diagram for two identical LIF-neurons with inhibitory coupling

Time constant for inhibitory synaps

May 27, 2005

Electrical coupling for spiking neuronsby gap junctional coupling

Out-of-phase stable In-phase stable

May 27, 2005

Phase-shift functionfor electrical coupling

effect of supra-threshold part of spike tends to synchronize activity

effect of sub-threshold part of spike tends to desynchronize activity

-70 mV

0 mV

+40 mV1.

2.1. 2.

May 27, 2005

Phase-shift functionfor electrical coupling

I=1.05

I=1.15

I=1.25

effect of supra-threshold part of spike tends to synchronize activity

effect of sub-threshold part of spike tends to desynchronize activity

effect of both components

May 27, 2005

Bifurcation diagram for two identical LIF-neurons with electrical coupling

May 27, 2005

Bifurcation diagram for two identical LIF-neurons with electrical coupling

May 27, 2005

If natural frequencies do not matchTime courses of hypathocyte x1 (solid line) and of x2 (dashed line) at P1=1.5 μM and P2=2.5 μM. (a) Harmonic locking of 1:3 (γCA=0.025 s-1); (b) harmonic lockingof 1:2 (γCA=0.05 s-1); (c) phase locking of 1:1 (γCA=0.09 s-1). (d) Devil’s staircase, a ratio N/M (where N is the spike number of x1 and M is the spike numberof x2) as a function of the coupling strength γCA at given IP3 level: P1=1.5 μM, P2=2.5 μM.

Wu et al., Biophys. Chem. 113, 2005Coupling strength

May 27, 2005

Bifurcation diagram for two identical LIF-neurons with inhibitory and electrical coupling

Inhibitory coupling only

Electrical coupling only

May 27, 2005

Electrical coupling in addition to synaptic (inhibitory) interactions

anti-phase, weak electrical coupling

in-phase , strong electrical coupling

no electrical coupling

anti-phase , weak electrical coupling

in-phase , strong electrical coupling

Brem & Rinzel, J. Neurophysiol. 91, 2004

May 27, 2005

Anti-phase and in-phase both stable

Stable in-phase

Stable anti-phase

Electrical coupling in addition to synaptic interactions

The stronger is the synaptic inhibition, the larger is the electrical coupling required to stabilize in-phase behavior

May 27, 2005

Summary

• Gap-junctions between two cells tend to synchronize the two oscillators

• synchronizing effect is stronger when there is a plateau phase in the action potential

May 27, 2005




May 27, 2005

What happens for two pacemaker cells with excitatory and gap-

junctional coupling ?

May 27, 2005

Two pacemaker cells

May 27, 2005

Synchronization of two oscillators

No coupling

Small conductance gap junction

Small conductance gap junction

May 27, 2005

Simple result for excitatory and electrical coupling

• Two pacemaker cells synchronize easily

May 27, 2005

Synchronization of activity in a network of cells

May 27, 2005

Network of NRK-cells

May 27, 2005

One pacemaker, surrounded by 6 followers

May 27, 2005

Two pacemaker cells

Ri

Rgap

RcellV

May 27, 2005

Network of NRK-cells

Ri

Rgap

Rcell

Rgap

Rcell

Rgap

Rcell

Experimental observation: a single pacemaker cell cannot initiate propagation of action potential firing

May 27, 2005

Resistance of gap-junction should not be too high and not

too low !

Ri

Rgap

Rcell

Rgap

Rcell

Rgap

Rcell

In the heart: Rcell is high !

May 27, 2005

Synchronization in a network of different coupled oscillators

May 27, 2005

Spontaneous oscillations and synchronization in NRK networks

Caer

Casyst

Membrane

potential

NRK cell with intracellular (IP3) oscillator and plasma membrane

Network

with NRK cells

Oscillations and synchronization

May 27, 2005

Standing problems• Cells are intrinsically stable, but become unstable

due to coupling in a network ?• Or: cells are unstable but synchronize in a network

to act as pacemakers for propagating activity ?• What is the role of electrical/gap-junctional

coupling and Ca-diffusion through gap junctions in propagation of action potential firing ?

• How to recognize pacemakers and followers ?• Pace-makers seem to “move” in a network

May 27, 2005

May 27, 2005

CacytBCacyt

B

IP3

Caex

GCaLClex

GCl(Ca)

KcytGKir

Glk

GCalk PMCA

CaerSERCA

IP3RJCalker

Complete model

GSOC

May 27, 2005

Further topics for study

• Compartimentalization:

– coupling of ER with cell membrane for store-operated channels

– discrete sources and sinks (stores)

– discrete channels : distance between channel clusters is larger than the diffusion length of free Ca2+

• stability of intracellular Ca2+ control

• relation between stochastic character of channel dynamics and deterministic periodic behavior of Ca-oscillations

May 27, 2005

References

• Falcke (2004) Reading the patterns in living cells —the physics of Ca2+ signaling. Advances in Physics, 53, 255–440

• Loewenstein, Yarom, Sompolinsky (2001) The generation of oscillations in networks of electrically coupled cells. PNAS 98, 8095-8100.

May 27, 2005


• CRAC kanaal• Ca2+ L type channel

• Cl(Ca) kanaal)(

11

1

))}10(0337.0{exp(0197.002.001.0

)24.5/)37exp((11

)10(035.0))9.5/)10(exp(1(01.0

)24.5/)15(exp(11

)(

)(1

)/(1

)()()(

2

)()()(

CaClClcyt

cytCaClCaCl

cytCa

h

m

LCaCaLCaLCa

CaCRACER

CRACCRAC

EVKCa

CaGI

Caw

V

Vh

VVm

Vm

EVwhmGI

EVKCa

CconvfluxI

May 27, 2005

Components of the model in the cell membrane

• CRAC channel

• Leakage into cytosol

• PMCA-pomp

)1000( cytlklk CaCJ

PMCAcyt

cytPMCAPMCA

KCa

CaCJ

)(1 Nernst

CaCRACER

CRACCRAC EVKCa

CJ

May 27, 2005

Overview of parameter values

MK

pFC

MK

nSG

nSG

nSG

nSG

VE

VE

VE

O

m

Cl

CaCl

LCa

K

leak

NernstCaCl

NernstLCa

Nernstleak

35

20

35

0.10

50.0

2.2

05.0

02.0

05.0

0

)(

)(

)(

)(

100000123.02

10

55.0

2.0

27.1

06.0

)(032.0

20

/96480

293

molKs kg m31.8

1

1

1

1

-1-1-22

convflux

MK

MsC

MK

MsC

sK

sMK

MTB

molCF

KT

R

CRAC

CRAC

PMCA

PMCA

off

on

MK

MK

MK

sK

MK

sC

sMC

sC

fIP

wIP

wIP

w

SERCA

lek

SERCA

IP

5.0

15

5

1

2.0

002.0

/6.0

10

3

32

31

1

1

13

for membrane for ER

May 27, 2005

Dynamics of IP3 regulated Ca2+ release

May 27, 2005

Ca-oscillations as a function of IP3

May 27, 2005

Oscillations in a large network

May 27, 2005

-150 -100 -50 0 50 100 1500

0.2

0.4

0.6

0.8

1

m(V)

-150 -100 -50 0 50 100 1500

2

4

6

8x 10

-3

m

(V)

m (

s)

-150 -100 -50 0 50 100 1500.2

0.4

0.6

0.8

1

1.2

h(V)

Vclamp (mV)-100 -50 0 50 1000

2

4

6

8

10

h(V)

Vclamp (mV)

h (

s)

Parameter fitting

May 27, 2005

Ca-action potentialstriggered by Ca-release from the ER

GCaL 20 mV

GCl(Ca) -20 mV

GKIR -70 mV

May 27, 2005

Phase diagram for closed-cell model

Sneyd et al., PNAS, 2004

May 27, 2005

Ca2+ is involved in the control of• Muscle contraction• memory storage• egg fertilization• enzyme secretion by acinar cell in pancreas• coordination of cell behavior in the liver• cell apoptosis• second messenger : coding and transfer of information

from cell membrane to nucleus• etc., etc., etc.

Yet, high cytosolic concentrations prohibit normal functioning of the cell. How can this be made compatibel ?

See Martin Falcke, Advances in Physics, 53, 2004

May 27, 2005

Different forms of Ca2+ oscillations

hepatocyte stimulated with norepinephrine

endothelial cellstimulated with histamine

sinusoidal oscillations in a parotid gland

May 27, 2005

Ca-dynamics

• Ca-oscillations in non-excitable cells

• Ca-inflow in excitable cells (action-potential generation) without intracellular Ca-oscillations.

• Ca-oscillations in cells with action-potentials and with IP3-mediated Ca-oscillations.

May 27, 2005

Overview• Summary of Hodgkin-Huxley model• Dynamics of Ca-oscillations and action

potentials• coupling between Ca-oscillations and action

potentials• Stability of Ca-dynamics in the cell• Propagation of electrical activity in network

of layers– oscillators as pacemakers which initiate propagation ?– instability due to coupling ?

May 27, 2005

V mV

0 mV

V mV

0 mV

IC

INa

Membrane voltage equation

-Cm dV/dt = gmax, Nam3h(V-Vna) + gmax, K n4 (V-VK ) + g leak(V-Vleak)

K

May 27, 2005

V (mV)

mmOpen Closedm

m

mProbability:

State:

(1-m)

Channel Open Probability:

dt

dm m)1( m m m

hhdtdh

hh )1(

Gating kinetics

m.m.m.h=m3h

mm

mm

mm

1

May 27, 2005

Actionpotential

May 27, 2005

Simplification of Hodgkin-HuxleyFast variables• membrane potential V• activation rate for Na+

m

Slow variables• activation rate for K+ n• inactivation rate for

Na+ h

-C dV/dt = gNam3h(V-Ena)+gKn4(V-EK)+gL(V-EL) + I

dm/dt = αm(1-m)-βmm

dh/dt = αh(1-h)-βhh

dn/dt = αn(1-n)-βnn

May 27, 2005

Phase diagram for the Morris-Lecar model

May 27, 2005

Phase diagram

May 27, 2005

Phase diagram of the Morris-Lecar model

May 27, 2005

Buffer dynamics

with

Kon = 0.032 (μMol s)-1

Koff = 0.06 s-1

CaBBCacyt

May 27, 2005

Phase-plane plot for membrane dynamics (Morris-Lecar model)

May 27, 2005

Ca L type channel activation (m∞) and inactivation (h ∞)

m∞

h ∞

V (mV)

May 27, 2005

The effect of Kon on the action potentialKon = 3.2 (μMol.s)-1Kon = 0.032 (μMol.s)-1

Kon = 0.32 (μMol.s)-1

Shorter APMore Ca buffered

Longer AP

All Ca buffered

Documents

May 27, 2005 Control of stability of intracellular Ca-oscillations and electrical activity in a network of coupled cells. Stan Gielen Dept. of Biophysics