The parallel‐conductance model for cell membranes
Example for squid axon
Let’s consider now the effect of variation
of K ions outside the cell membrane
A. Increasing the external K+ concentration
makes the resting membrane potential more
positive.
B. Relationship between resting membrane
potential and external K+ concentration.
This experiments show that the inside‐negative
resting potential arises because (1) the
membrane of the resting neuron is more
permeable to K+ than to other ions, and (2) there
is more K+ inside the neuron than outside .
The selective permeability to K+ is caused by K+‐permeable membrane channel that are open in
resting neurons; and the large K+ concentration gradient is produced by membrane transporters
that selectively accumulate K+ within neurons.
Biosensors and Bioelectronics II (SS 2011)
31
Jose A. Garrido | garrido@wsi.tum.de
The ionic basis of action potentials
What causes the membrane potential of a neuron to depolarize during an action potential?
Hodgkin and Katz examined the role of Na+ in generating the action potential by changing the Na+
concentration in the external medium
Lowering the external Na+ concentration reduces both the rate of
rise of the action potential and its peak amplitude.
While the resting membrane
potential is almost independent of
the Na+ concentration, the
membrane
becomes
extraordinarily permeable to Na+
during the rising phase .
The temporary increase in Na+
permeability results from the
opening of Na+‐selective channels
that are closed in the resting state.
Biosensors and Bioelectronics II (SS 2011)
32
Jose A. Garrido | garrido@wsi.tum.de
Voltage‐dependent membrane permeability
I m   I j I C  I Na  I K  I Cl
The total membrane current depends on the membrane voltage
j
I C  Cm
dV
dt

eq
I Na  GNa V  VNa


I K  GK V  VKeq


I Cl  GCl V  VCleq

Neuroscience, 4th Edition, Figure 3.1
Current flow across a squid axon membrane during a voltage clamp experiment
Hiperpolarization of the membrane (from the resting at ‐65 mV to ‐135 mV)
capacitive currents
Depolarization of the membrane (from the resting at ‐65 mV to 0 mV) results in (1) fast capacitive
current transient followed by a (2) rapid rising inward current (positive charge entering the cell),
which gives rise to a slowly rising, delayed outward current.
voltage‐sensitive permeability
Biosensors and Bioelectronics II (SS 2011)
33
Jose A. Garrido | garrido@wsi.tum.de
Voltage‐dependent membrane permeability
Neuroscience, 4th Edition, Figure 3.2
Two types of voltage‐dependent ionic current
Current produced by membrane depolarizations to several different potentials
The early current first increases with potential and then decreases; at potentials more
positive than about +55 mV, this current changes polarity.
The late current increases monotonically with increasingly positive membrane potentials.
Biosensors and Bioelectronics II (SS 2011)
34
Jose A. Garrido | garrido@wsi.tum.de
Voltage‐dependent membrane permeability
The early inward current is carried by the entry of Na+ into the
cell. At +52 mV, no Na+ flux occurs, as this is approximately the
equilibrium potential for Na+ ions (in squid neurons).
Pharmacological separation of Na+ and K+ currents
Neuroscience, 4th Edition, Figure 3.5
Changing the membrane potential
to a value more positive than the
resting potential produces an early
influx of Na+ (inward current) into
the neuron, followed by a delayed
efflux of K+ (outward current)
Current amplitude versus voltage
membrane. The early current curve
represents the amplitude of the peak;
the late current curve corresponds to the
saturation value of the current.
Biosensors and Bioelectronics II (SS 2011)
35
Jose A. Garrido | garrido@wsi.tum.de
Voltage‐dependent membrane permeability
Voltage‐dependent membrane conductances
Gj 
Ij
V  V 
eq
j
Both conductances require
some time to activate, or
turn on.
The K+ conductance has a
pronounced delay, while
the Na+ conductance
reaches its maximum
more rapidly.
Depolarization
causes
both the activation and
deactivation of the Na+
conductance.
Both peak Na+ and steady‐state K+ conductances increase as the Vm becomes more positive
The activation of both conductances, and the rate of deactivation of the Na+ conductance
occur more rapidly with larger depolarization potentials
Biosensors and Bioelectronics II (SS 2011)
36
Jose A. Garrido | garrido@wsi.tum.de
Voltage‐dependent membrane permeability
Neuroscience, 4th Edition, Figure 3.7
Voltage‐dependent membrane conductances
Both peak Na+ and steady‐state K+ conductances increase as the Vm becomes more positive
Biosensors and Bioelectronics II (SS 2011)
37
Jose A. Garrido | garrido@wsi.tum.de
Single‐channel (Na+) conductance
Using the patch‐clamp technique in the inside‐out configuration, it is
possible to measure single‐channel currents.
Successive recordings in the same channel reveal minuscule
microscopic (inwards) currents (1‐2 pA), in which the channel randomly
switches to and from an open or closed state.
The sum of many recordings shows that most channels open in the
initial 1‐2 ms following depolarization, after which the probability of
channel openings diminishes (channel inactivation).
single‐channel recordings in a Na+
channel
The probability of channel
opening increases with the
applied membrane potential, in
a similar way than the
membrane conductance.
The macroscopic current measurement
(whole‐cell config.) shows the close
correlation between the time courses of
microscopic and macroscopic Na+ currents.
Biosensors and Bioelectronics II (SS 2011)
38
Jose A. Garrido | garrido@wsi.tum.de
Single‐channel (K+) conductance
Single‐channel recordings in a K+ channel
Neuroscience, 4th Edition, Figure 4.2
During the depolarization stimulation, the probability of channel opening does not decrease with time, i.e., there is not inactivation of the K+ channels (in contrast to the Na+ channels) Biosensors and Bioelectronics II (SS 2011)
39
Jose A. Garrido | garrido@wsi.tum.de
Functional states of Na+ and K+ channels
Biosensors and Bioelectronics II (SS 2011)
40
Jose A. Garrido | garrido@wsi.tum.de
The Hodgkin‐Huxley membrane model
Hodgkin and Huxley proposed a mathematical model to fit
the measurements of the channel conductance of squid
giant axons.
For the potassium channel, it was assumed that it would
open only if four independent subunits of the channel (see
slide 10, K+ ion‐channel structure ) had moved from a closed
to an open position.
probability of 4

p

n
+
K
the K channel to be open
with n the probability
of each subunit to be
open n ≡ f(t,Vm)
The movement of each subunit from close to open, or open
to close is assumed to be described by first order kinetics,
with rate constants n and n, respectively.
dn
  n (1  n)   n n
dt
n
→ open
close ←
with the rate constants depending only on Vm, which is constant
for voltage‐clamp experiments
n(t )  n  (n  n0 )e  t / n
n
Biosensors and Bioelectronics II (SS 2011)
 n  ( n   n ) 1
n   n ( n   n ) 1
41
Jose A. Garrido | garrido@wsi.tum.de
The Hodgkin‐Huxley membrane model
Then, the total K+ conductance can be calculated from the fraction of
open channels (n4) and the conductance when all channels are open:
GK (t , Vm )  GK n 4 (t , Vm )
GK  total conductance when all channels are open
The dependence of the rate constants on the transmembrane
potential was derived from the fitting to the experimental data.



   1
1.5




-1
  10  Vm  V rest
exp
10
 
n, n (msec )
n 
0.01(10  Vm  V rest )
n
→ open
close ←
n

  Vm  V rest
 n  0.125 exp
80

 


n
1.0
0.5
n
0.0
‐20
0
20
40
Vm-V
Biosensors and Bioelectronics II (SS 2011)
42
60
rest
80
100
120
140
(mV)
Jose A. Garrido | garrido@wsi.tum.de
The Hodgkin‐Huxley membrane model
GK (t , Vm )  GK n 4 (t , Vm )
Vm-V rest= 100mV
0
2
4
6
n(t )  n  (n  n0 )e  t / n
8
 n  ( n   n ) 1
n   n ( n   n ) 1
Vm-V rest= 60mV
0
2
4
6
8
Vm-V rest= 26mV
0
2
4
time (msec)
Biosensors and Bioelectronics II (SS 2011)
43
6
8
n 


0.01(10  Vm  V rest )

  10  Vm  V rest
exp
10
 
   1





  Vm  V rest
 n  0.125 exp
80

Jose A. Garrido | garrido@wsi.tum.de
 


The Hodgkin‐Huxley membrane model
p K (Vm )  n4 (Vm )
1.0
probability of K+ channel opening
Neuroscience, 4th Edition, Figure 4.2
0.5
0.0
0
20
40
Vm-V
Biosensors and Bioelectronics II (SS 2011)
44
60
80
100
120
rest
(mV)
Jose A. Garrido | garrido@wsi.tum.de
The Hodgkin‐Huxley membrane model
For sodium channels, which show both activation and inactivation processes,
the probability of channel opening was assumed to be controlled by two
different types of subunits.
probability of 3
the Na+ channel  p Na  m h
to be open
with m, and h the so‐called
activation
and
inactivation
parameters, respectively.
As with K+ channels, m and h correspond to the probability of certain
subunits to be open. There are three subunits with open probability m and
one with open probability h.
Both parameters satisfy first‐order differential equations:
dh
  h (1  h)   h h
dt
for m‐type subunits
for h‐type subunits
m
h
→ open
close ←
→ open
close ←
m
Biosensors and Bioelectronics II (SS 2011)
dm
  m (1  m)   m m
dt
h
45
Jose A. Garrido | garrido@wsi.tum.de
The Hodgkin‐Huxley membrane model
dh
  h (1  h)   h h
dt
dm
  m (1  m)   m m
dt
As before, the solution of this equations leads to
m(t )  m  (m  m0 )e  t / m
h(t )  h  (h  h0 )e  t / h
 m  ( m   m ) 1
 h  ( h   h ) 1
m   m ( m   m ) 1
h   h ( h   h ) 1
Then, the total Na+ conductance can be calculated from the fraction of open
channels (m3h) and the conductance when all channels are open:
GNa (t , Vm )  GNa m 3 (t , Vm )h(t , Vm )
GNa  total conductance when all channels are open
The dependence of the rate constants on the transmembrane potential
was derived from the fitting to the experimental data.
m 
0.1(25  Vm )
exp0.1(25  Vm )   1
  Vm 

 18 
 m  4 exp
Biosensors and Bioelectronics II (SS 2011)
  Vm 

20


 h  0.07 exp
h 
1
exp0.1(30  Vm )   1
46
Jose A. Garrido | garrido@wsi.tum.de
The Hodgkin‐Huxley membrane model
10
m‐type subunits
→ open
close ←
-1
m, m (msec )
m
m
8
m
6
4
m 
m
0.1(25  Vm )
exp0.1(25  Vm )   1
  Vm 

 18 
 m  4 exp
2
0
‐20
0
20
40
60
Vm-V
80
100
120
140
rest
(mV)
h‐type subunits
1.0
h
h
→ open
close ←
-1
h, h (msec )
0.8
h
0.6
  Vm 

 20 
 h  0.07 exp
0.4
0.2
0.0
‐20
h
h 
0
20
40
60
Vm-V
80
100
120
140
1
exp0.1(30  Vm )   1
rest
(mV)
Biosensors and Bioelectronics II (SS 2011)
47
Jose A. Garrido | garrido@wsi.tum.de
The Hodgkin‐Huxley membrane model
GNa  GNa m 3 h
Vm-V rest= 100mV
0
1
2
3
m(t )  m  (m  m0 )e  t / m
 m  ( m   m ) 1
4
m   m ( m   m ) 1
h(t )  h  (h  h0 )e  t / h
Vm-V rest= 60mV
0
2
 h  ( h   h ) 1
4
h   h ( h   h ) 1
time (msec)
m 
0.1(25  Vm )
exp0.1(25  Vm )   1
  Vm 
 m  4 exp

 18 
Biosensors and Bioelectronics II (SS 2011)
48
  Vm 

20


 h  0.07 exp
h 
1
exp0.1(30  Vm )   1
Jose A. Garrido | garrido@wsi.tum.de