Celbiofysica voor MLW studenten

1
Celbiofysica voor MLW studenten

• Onderdeel Biofysica

www.mbfys.ru.nl/stan/
2
Onderdelen Biofysica

• Het passieve membraan
• Hodgkin-Huxley model voor actief membraan
• Intracellulaire calcium oscillaties
• Werkcolleges en computer simulaties

3

• Doelstellingen Cel-biofysica voor MLW
• Inzicht geven in passieve electrische
  eigenschappen van celmembraan
• Inzicht geven in de fysische principes van
  ion-transport door een membraan.
• Inzicht geven in spanningsafhankelijke dynamiek
  van kanalen in membraan, die een rol spelen bij
  functioneren van de levende cel.
• Inzicht geven in dynamiek van calcium
  oscillaties voor intracellulaire communicatie
• Gebruik van computer simulaties om inzicht te
  geven in fysiologische mechanismen van membraan
  transport.

4
Various types of neurons
5
A cartoon of the neuron
6
How does a membrane of a neuron look like ?

• Layer with phospholipids with polar head at
  outside and hydrophobic tails inside.
• Large proteins in phospholipid layer

7
A slightly more realistic view of a membrane.
8
Model neuron
9
(No Transcript)
10

• Membraan biofysica
• Cel krijgt input van omgeving via
• synapsen, die de permeabiliteit van membraan
  veranderen, waardoor
• ionen door de membraan kunnen migreren en
• de interne potentiaal van een cel verandert.

11
Electricity and charges
12
Bio-Electriciteit

• 2 gelijke electrische ladingen stoten elkaar af
• 2 verschillende electrische ladingen trekken
  elkaar aan

13
Electric field
Q charge (Coulomb) r distance E electric
field V potential (Volt)
r
Q
E 1/(4pe) Q/r2 V 1/(4pe) Q/r
14
Force on charge in electric field
Q charge (Coulomb) r distance E electric
field V potential (Volt) e di-electric
constant
Q2
r
Q1
E 1/(4pe) Q1 /r2 F Q2 E 1/(4pe) Q1 Q2 /r2
15
Model neuron
16
Ohms law for a resistor
V Potential difference (Volt) R Resistance I
Current (Ampere) g 1/R conductance VI.R I
V/RgV
R
V
I
17
Voorbeeld

I
Batterij
6V
R3 Ohm
-
I V/R 6/3 2 Ampere
18
Serie van weerstanden
RR1R2R3 1/g1/g1 1/g2 1/g3 1/R 1/R1
1/R2 1/R3 g g1 g2 g3
19
Voorbeeld 1
Een axon van de Reuze Inktvis heeft straal van
0.25 10-3 m en een specifieke membraan
geleidbaarheid van 1.4 10-3 S/cm2. Hoe groot is
de geleidbaarheid van een stukje axon met lengte
1 cm ? Oppervlak van het axon A 2pr l 2p
(0.25 10-1 cm)(1 cm) 0.16 cm2 G sA 1.4
10-3 A 1.4 10-3 x 0.16 2.2 10-4 (S) Weerstand
R 1/(2.2) 104 (Ohm)
20
Voorbeeld 2
Geleidbaarheid van een stuk axoplasma van de
reuze inktvis met lengte 1 cm is 3.3 10-2
S/cm. Hoe groot is de geleidbaarheid van een stuk
axon van lengte 10 cm en met straal 0.025 cm
? gsA/L 3.3 10-2 (S/cm) x p (0.025)2 (cm2)/(10
cm) 6.5 10-6 (S) R 1/g 1/(6.5) 106 (Ohm)
21
Capacitor
C eA/d
22
Capacitor
Q Charge (Coulomb) V Potential difference I
current (Ampere) dQ/dt C Capacitance (Farad)
I
V
C
QCV I dQ/dt C dV/dt
23
Current through an ion channel
24
Electrical equivalent
Biology
R
I1
I2
I
I I1 I2
25
Current through membrane
R
I I1 I2 V/R C dV/dt
I1
I2
I
I I1 I2
26
Current through an ion-channel
0 mV
C
I
I
V mV
Ohms law
Q C V I dQ/dt C dV/dt
Q charge C capacitance
Conductance G1/R
27
Current through Resistor and Capacitance
V
I
I V/R
R
V
I
C
IC dV/dt
28
Resistor and Capacitance in parallel
I
I1
I2
V
I
I2
I1
time
29
Resistor and Capacitance in parallel
I I1 I2 V/R C dV/dt Als I0 voor tlt0 en
II0 voor tgt0 V(t) RI0 (1 - e - t / RC)
I
I1
I2
I
V
time
30
Stroom door membraan
Geleidbaarheid van membraan van 1 cm lengte 2 p
rsm Stroom in lengte richting door membraan
gradient van potentiaal x geleidbaarheid per
lengte eenheid van het axoplasma Il -p r2 s
dV/dx Stroom door het membraan Im - dIl /dx p
r2 s d2 V/dx2 Maar ook Im 2p r sm V (V
buitenmembraan 0) 2p r sm V p r2 s d2 V/dx2 ?

31

• Electrical potential within the cell is at - 80
  mV with respect to outside.
• Across the membrane, there is a potential
  difference of -80 mV.

32
The passive membrane

• As a first step to understand the electrical
  properties of the membrane, we will consider the
  membrane as a passive system with diffusion of
  ions due to
• different concentrations of ions at both sides
  of the membrane
• potential difference across the membrane

33
Nernst equation

• Gives a relation between
• concentration difference of ions
• potential difference
• in equilibrium.

34
Passive diffusion
Flux is defined as the current per unit
area. Flux is given by J P( C I - C II
) with Ci is concentration in medium I P is
permeability
35
Osmose diffusion Balance between concentration
and pressure gradient
Osmotic pressure difference ?p is defined by ?p
RT ?C with T temperature ?C
difference in concentration R gas
constante (8.3 J/(K mol))
36
Flux tgv concentratie verschil en potentiaal
gradient
Concentratie verschil
Potentiaal gradient
37
Flux tgv concentratie verschil en potentiaal
gradient op positie x in het membraan
C1 V1
C2 V2
Tgv potentiaal gradient
tgv concentratie gradient
38
Flux tgv concentratie verschil en potentiaal
gradient in evenwicht
C1 V1
C2 V2
39
Nernst Equation
The Nernst equation relates the potential
difference to the concentration difference in
equilibrium Ci / Co exp - Z e (Vi -Vo )/kT
or Vi - Vo (kT/Ze) ln (Co/Ci) with e charge
electron Z valence of ion k Boltzmann
constant T temperature Ci (Co) concentration
inside (outside) membrane
40
Example of Nernst equation
t0
t gtgt 0
Na 100 Cl 100
Na 10 Cl 10
Na 55 Cl 55
Na 55 Cl 55
?V 0
?V 0
If both Na and Cl- can migrate through the
membrane
41
Example of Nernst equation
t0
t gtgt 0
R 50 Na 50 Cl- 100
Na 100 Cl- 100
R 50 Na 64 Cl- 110
Na 86 Cl- 90
?V 0
?V 0
If both Na and Cl, but not R can migrate through
the membrane.
42
Nernst equation for squid axon
43
Caveats regarding Nernst equation

• considers only a single ion.
• If multiple ions are involved, it assumes
  equal permeability for all ions
• Applies only to passive transport
• ions migrate independently of each other

44
Goldman/Hodgkin/Katz equation
The Nernst equation relates the potential
difference to the concentration difference in
equilibrium for a single neuron. When several
ions are involved, we obtain for equilibrium
Pk KoPNaNaoPClCli ?V (RT/F) ln
--------------------------------- Pk
KiPNaNaiPClClo with Pi permeability of
ion i Ki/o concentration inside/outside F
Faraday constant T temperature ?V
potential difference across membrane
45
Current through an ion channel
46
Current through an ion-channel
0 mV
C
I
I
V mV
Ohms law
Q C V I dQ/dt C dV/dt
Q charge C capacitance
Conductance G1/R
47
Current through Resistor and Capacitance
V
I
I V/R
R
V
I
C
IC dV/dt
48
Resistor and Capacitance in parallel
I
I1
I2
V
I
I2
I1
time
49
Resistor and Capacitance in parallel
I I1 I2 V/R C dV/dt Als I0 voor tlt0 en
II0 voor tgt0 V(t) RI0 (1 - e - t / RC)
I
I1
I2
I
V
time
50
Electrical scheme of passive membrane
51
The active membrane

• The membrane is not passive, but has active
  properties
• Active Na /K pump

52
Schematic overview of the active membrane
53
Current through an ion-channel
0 mV
R
I
I
Vion
V mV
Ohms law
I (?V - Vion )/R G (?V - Vion )
Conductance G1/R
54
Hodgkin Huxley Current through an ion-channel
0 mV
R
I
I
Vion
V mV
Ohms law
Conductance G1/R
Conductance G is a product of maximal conductance
gCa and the fraction of open channels m3h
55
Gating kinetics
State
56
Gating kinetics
Open
State
57
Gating kinetics
?m
Open
Closed
State
?m
m
(1-m)
Probability
?m
?m
58
Gating kinetics
?m
Open
Closed
State
?m
m
(1-m)
Probability
?m
?m
Steady-state oplossing
59
Gating kinetics
?m
Open
Closed
State
?m
m
(1-m)
Probability
?m
?m
Steady-state oplossing
Oplossing als functie van de tijd na stap V
met
60
Gating kinetics
?m
?m
?m
Open
Closed
State
?m
m
(1-m)
Probability
?m
?m
Channel Open Probability
61
Gating kinetics
?m
?m
?m
Open
Closed
State
?m
m
(1-m)
Probability
?m
?m
Merk op wat er gebeurt als en beide groter
worden !!
62
Parameter fitting (2)
63
Parameter fitting (2)
64
Voltage dependence
65
Gating kinetics
?m
Open
Closed
State
?m
m
(1-m)
Probability
?m
?m
m(t) m8 (m0 - m0)e-t/t
h(t) h8(h0 - h8)e-t/t
Channel Open Probability
m3h
66
t0
m(t) m8 (m0 - m0)e-t/t
h(t) h8(h0 - h8)e-t/t
Channel Open Probability
m3h
67
Na stap-vormig increment in V
m(t) m8 (m0 - m0)e-t/t
h(t) h8(h0 - h8)e-t/t
Channel Open Probability
m3h
68
m(t) m8 (m0 - m0)e-t/t
h(t) h8(h0 - h8)e-t/t
Channel Open Probability
m3h
69
Hodgkin Huxley Current through an ion-channel
0 mV
R
I
I
Vion
V mV
Ohms law
Conductance G1/R
Conductance G is a product of maximal conductance
gCa and the fraction of open channels m3h
gmax m3h ?V
The driving force DV is the difference between
the membrane potential V and the Nernstpotential
Vion for the ion.
gmax m3h (V-Vion)
70
Membrane voltage equation
0 mV
0 mV
INa
IC
V mV
V mV
Kirchoffs law
INa gmax, Nam3h(V- VNa)
-Cm dV/dt gmax, Nam3h(V-Vna) gmax, K n4 (V-VK
) g leak(V-Vna)
71
Actionpotential
72
(No Transcript)
73
(No Transcript)
74
(No Transcript)
75
How to measure membrane currents ?
76
Patch-clamp recording
A Patch-clamp set-up B Bursts of outward
current (1 to 5 pA) when channel is open. Pulse
amplitude is constant. Duration varies.
77
Channel specific gating
Each channel is specifically sensitive to
specific neurotransmitters, which affect the
state (open/closed) of the channel.
78
Current-Voltage relation for single-channel
currents.
79
Ion currents through membrane
Single-channel currents from foetal (A, C) and
adult (B, D) channels from bovine muscle (left)
and xenopus oocytes.
80
Various patching techniques
81
Hodgkin-Huxley model

• Fast 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 am(1-m)-ßmm dh/dt ah(1-h)-ßhh dn/dt
an(1-n)-ßnn
82
Actionpotential
-C dV/dt gNam3h(V-Ena)gKn4(V-EK)gL(V-EL)
I dm/dt am(1-m)-ßmm dh/dt ah(1-h)-ßhh dn/dt
an(1-n)-ßnn
83
Intracellulaire Calcium oscillaties
84
Intra-cellular signaling
External input from other cells
85
(No Transcript)
86
(No Transcript)
87
Cells reveal non-linear behavior

• Cell mitosis
• Gene-expression and protein synthesis
• Folding/unfolding of proteins (hormone secretion)

Discrete states Once started complete it
88
NRK-cells exotic ?

• No, they are not
• Smooth-muscle cells (gastrointestinal)
• lymphatic
• vascular tissues
• heart pacemaker cells.
• urethral
• Hormone producing cells in the brain

89
Growth states of NRK fibroblasts
90
Growth states of NRK fibroblasts
91
Growth states of NRK fibroblasts
Membrane potential
92
Intracellular Ca2-oscillations

• Ca2 is a second messenger

93
(No Transcript)
94
Membrane oscillator
CaER
(1000 µM)
(0.1 µM)
(1000 µM)
95
The basic elements are the following

• PMCA pump
• Ca L type channel
• Cl(Ca) channel
• SOC channel
• Leak channel
• Kir channel
• Ca-buffer in the cytosol

96
Components of the model for the NRK Membrane

• Lek door het membraan
• Kalium kanaal
• PMCA-pomp

97
Components of the model for the NRK Membrane

• SOC kanaal
• Ca L type kanaal
• Cl(Ca) kanaal

98
Buffer dynamics

• with
• Kon 0.032 (microMol s)-1
• Koff 0.06 s-1

99
Steady state behavior without current

• Upper left panel Cacyt (red trace), BCa
  (buffered complex, blue trace) and J_PMCA (green
  trace)
• Upper right panel VM (blue trace)
• lower left panel ICl (red trace), ICa (black
  trace), IK (green trace), Ileak(blue trace),
  ICRAC (cyan trace) and Ipulse (magenta trace)

100
Current clamp Ipulse3 pA
101
Current clamp Ipulse6 pA

• When we current clamp, the activation gate of the
  Ca L type opens, giving rise to a constant inflow
  of Ca through the Ca L type channel.
• As a consequence, an action potential will be
  generated

102
Current clamp Ipulse6 pA
103
Current clamp Ipulse6 pA
Action potential
Cacyt
Plateau due to Nernst potential of Ca-dependent
Cl-channel
Inflow of Ca ions
Pump current
104
Effect van aanwezigheid buffer

• Removing the buffer does affect
• Cacyt (smaller concentrations with buffer)
• AP (smaller AP with buffer)

105
Current clamp Ipulse6 pA
106
Current clamp Ipulse6 pA without buffer
107
Model for IP3-oscillator
108
The dynamics of Ca in the ER and cytosol by
transport through the IP3 receptor and by the
cell membrane, excluding the L-type Ca-channel
and K- or Cl ion channels 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
• SOC-channel
• Ca-buffer in the cytosol

109
Components of the model
Glek
CaER
SERCA
Jlek
PLC
BCacyt
ATP
IP3
Cacyt
B
IP3 receptor
JCRAC
Jlk
ATP
ATP
PMCA
110
Components of the model for the IP3-oscillator

• IP3-receptor
• Leakage from ER
• SERCA-pomp

111
Components of the model in the cell membrane

• SOC channel
• Leakage into cytosol
• PMCA-pomp

112
steady-state behavior

• Without IP3, the steady-state is easily found by
  solving the two equations, with two unknown
  variables Cacyt and CaER

This gives a single, stable solution for Cacyt
and CaER
Vergelijk y2x1 y5x-2 (oplossing x1 y3)
113
Steady state behavior without IP3

• Upper left panel Cacyt (red trace) and BCa
  (buffered complex, blue trace)
• Upper right panel CaER
• lower left panel JSERCA (red trace), Jlek (blue
  trace) and JIP3 (green trace)
• lower left panel JPMCA (red trace), Jlk (blue
  trace), and JSOC (green trace).

It may take a long time, but the system always
converges to a stable state, irrespective of
various parameters, as long as JCRAC is large
enough .
114
Concentration IP35

• When the concentration of IP3 increases above
  zero, the activation gate of the IP3 receptor
  opens, giving rise to a constant inflow of Ca
  through the IP3 receptor.
• As a consequence, CaER decreases relative to the
  situation for IP30 .
• Note that near the end we have JPMCA JSOC Jlk
  and JSERCAJlek JIP3
• Note that JIP3 does not oscillate the system
  goes to a stable steady state with a steady state
  value for activation f and inactivation w of the
  IP3 receptor.
• A similar result is found for IP3 8

115
IP3 concentration 5

• Upper left panel Cacyt (red trace) and BCa
  (buffered complex, blue trace)
• Upper right panel CaER
• lower left panel JSERCA (red trace), Jlek (blue
  trace) and JIP3 (green trace)
• lower left panel JPMCA (red trace), Jlk (blue
  trace), and JSOC (green trace).

116
Concentration IP38

• Upper left panel Cacyt (red trace) and BCa
  (buffered complex, blue trace)
• Upper right panel CaER
• lower left panel JSERCA (red trace), Jlek (blue
  trace) and JIP3 (green trace)
• lower left panel JPMCA (red trace), Jlk (blue
  trace), and JSOC (green trace).

117
IP3 oscillations

• When IP3 concentration is increased to 10
  we obtain oscillations of the IP3-receptor. A
  more detailed analytical analysis is necessary to
  understand why f and w reach steady state values
  (like for IP3 is 0,5 or 8) or start oscillating.
• Averaged over time we find that JPMCA JSOC Jlk
  and JSERCAJlek JIP3
• Note that the mean CaER concentration decreases
  to smaller values for higher IP3 concentrations.
• Raising the IP3 concentration to higher values
  increases the oscillation frequency.

118
Concentration IP310
119
Concentration IP312
120
What happens at higher IP3 concentrations ?

• Increasing the IP3- concentration to higher
  values increases the oscillation frequency.
• Notice that the first peak of JIP3 is large, but
  subsequent flux peak values are much smaller.
  This can be understood from the following After
  the first Ca-release from the ER, the SERCA pump
  starts reloading the ER. Since the next
  IP3-release comes earlier for higher IP3
  concentrations, the Cacyt is higher at the
  following peak. As a consequence CaER-Cacyt is
  smaller giving rise to smaller IP3 outflow.
  Moreover, the inactivation closes faster due to
  the higher IP3 concentration (see equations on
  sheet 4).

121
Concentration IP320
122
Components of the model for the IP3-oscillator

• IP3-receptor
• Leakage from ER
• SERCA-pomp

123
Concentration IP350
124
Dynamics of IP3 receptor
nucleus
125
Coupling between excitable membrane and
intracellular calcium oscillations
126
Coupling between excitable membrane and
intracellular calcium oscillations
127
Stability analysis of IP3 receptor
nucleus
128
Stability analysis for membrane and IP3-receptor
129
Poincaré map
The map M Cn ?Cn with can be written in
diagonal form with eigenvalues or Floquet
multipliers with Stability requires
130
Stability and hysteresis
131
Stability and hysteresis
132
Hysteresis as a function of SOC-conductance
133
Summary

• Coupling of two relatively simple systems gives a
  complex system with multi-stability and
  hysteresis
• Multi-stability provides several states
  controlled by a few parameters.
• Hysteresis ensures that a process is completed in
  the presence of external changes.

134
Wand ring
PGF2a
135
nifedipine
Nifedipine blocks the L-type Ca-channel and
thereby eliminates propagation of electrical
activity and Ca-wave propagation
136
(No Transcript)
137
(No Transcript)
138
(No Transcript)
139
(No Transcript)
140
(No Transcript)
141
(No Transcript)
142
Components of the model for the NRK Membrane

• Lek door het membraan
• Kalium kanaal
• PMCA-pomp

143
Hysteresis diagram for cell membrane
Caer
GCl(Ca)
GCaL
Caex
144
Chemical potential
Chemical potential In equilibrium