Transcript Document

Control of Cell Volume and
Membrane Potential
Basic reference: Keener and Sneyd, Mathematical Physiology
Basic problem
• The cell is full of stuff. Proteins, ions, fats, etc.
• Ordinarily, these would cause huge osmotic pressures, sucking
water into the cell.
• The cell membrane has no structural strength, and the cell would
burst.
Basic solution
• Cells carefully regulate their intracellular ionic concentrations, to
ensure that no osmotic pressures arise
• As a consequence, the major ions Na+, K+, Cl- and Ca2+ have different
concentrations in the extracellular and intracellular environments.
• And thus a voltage difference arises across the cell membrane.
• Essentially two different kinds of cells: excitable and nonexcitable.
• All cells have a resting membrane potential, but only excitable cells
modulate it actively.
The cell at steady state
3 Na+
We need to model
• pumps
• ionic currents
• osmotic forces
2 K+
Cl-
Ca2+
Typical ionic concentrations
Squid Giant Axon
Frog Sartorius
Muscle
Human Red Blood
Cell
Intracellular
Na+
50
13
19
K+
397
138
136
Cl-
40
3
78
Na+
437
110
155
K+
20
2.5
5
Cl-
556
90
112
Extracellular
Active pumping
• Clearly, the action of the pumps is crucial for the maintenance of
ionic concentration differences
• Many different kinds of pumps. Some use ATP as an energy source
to pump against a gradient, others use a gradient of one ion to pump
another ion against its gradient.
• A huge proportion of all the energy intake of a human is devoted to
the operation of the ionic pumps.
Na+-K+ ATPase
Na+-K+ ATPase
Outside the cell
K+
Na+
K+•C•P
Na+•C•P
Step 2
ADP
Step 1
Step 3
P
ATP
Step 4
Na+•C
K+•C
K+
Na+
Inside the cell
Osmosis
P1
water
At equilibrium:
P2
water +
Solvent
(conc. c)
P1  kcT  P2
Note: equilibrium only. No information about the flow.
The Nernst equation
[S]i=[S’]i
[S]e=[S’]e
Vi
Ve
Permeable to S,
not S’
RT [S]e 
Vi  Ve 
ln

F [S]i 
(The Nernst potential)
Note: equilibrium only. Tells us nothing about the current. In addition,
there is very little actual ion transfer from side to side.

We'll discuss the multi-ion case later.
Only very little ion transfer
spherical cell - radius 25 mm
surface area - 8 x 10-5 cm2
total capacitance - 8 x 10-5 mF (membrance capacitance is about 1 mF/cm2)
If the potential difference is -70 mV, this gives a total excess charge on the cell
membrane of about 5 x 10-12 C.
Since Faraday's constant, F, is 9.649 x 104 C/mole, this charge is equivalent to
about 5 x 10-15 moles.
But, the cell volume is about 65 x 10-9 litres, which, with an internal K+
concentration of 100 mM, gives about 6.5 x 10-9 moles of K+.
So, the excess charge corresponds to about 1 millionth of the background K+
concentration.
Electrical circuit model of cell
membrane
outside
C
Iionic
How to model
this
inside
dV
C
 Iionic  0
dt
Vi Ve  V
C dV/dt
How to model Iionic
• Many different possible models of Iionic
• Constant field assumption gives the Goldman-Hodgkin-Katz model
• The PNP equations can derive expressions from first principles
(Eisenberg and others)
• Barrier models, binding models, saturating models, etc etc.
• Hodgkin and Huxley in their famous paper used a simple linear model
• Ultimately, the best choice of model is determined by experimental
measurements of the I-V curve.
Two common current models
INa  gNa (V  VNa )
Linear model
INa
 
[Na ]  [Na ] exp VF
i
e
F 
RT
 PNa
V
VF
RT 
1
exp

RT
2
 




GHK model
These are the two most common current models. Note how they
both have the same reversal potential, as they must.
(Crucial fact: In electrically excitable cells gNa (or PNa) are not
constant, but are functions of voltage and time. More on this
later.)
Electrodiffusion: deriving current
models
cell membrane
Outside
Inside
[S 1+] = [S 2-] = c i
Poisson equation and
electrodiffusion
d 2
 2 (c1  c 2 ), 2  st uff  L2
2
dx
dc
F d 
J1  D1 1 
c1 
dx RT dx 
dc
F
d 
J 2  D2  2 
c 2 
 dx RT dx 
S1
[S 1+] = [S 2-] = c e
S2
x=0
x=L
 (0) = V
f (L) = 0
Boundary conditions
c1(0)  c i ,
c 2 (0)  c i ,
c1(L)  c e
c 2 (L)  c e
 (0)  V,
 (L)  0

The short-channel limit
If the channel is short, then L ~ 0 and so  ~ 0.
d 2
d
T hen 2  0, which implies that the electric field, , is constant through the membrane.
dx
dx
d
dc1
v 
 vc1  J1
dx
dx
c i  c e ev
 J1  v
1 ev

VF 
D1F 2 c i  c e exp RT 
 I1 
V
LRT  1 exp VF 


RT
 
 
This is the Goldman-Hodgkin-Katz equation.

Note: a short channel implies independence of ion movement through the
channel.

The long-channel limit
If the channel is long, then 1/L ~ 0 and so 1/ ~ 0.
1 d 2
T hen 2 2  c1  c 2, which implies thatc1  c 2 through the membrane.
 dx
dc
c1  c 2  2 1  J1  J2
dx
 c1  c i  (c e  c i )x
v c i  c i  
    ln   1 x
v1  nondimensional Nernst potential of ion 1
v1 c e  c e  
c c
 J1  e i (v  v1 )
v1
This is the linear I-V curve.
 The independence principle is not satisfied, so no independent movement of
ions through the channel. Not surprising in a long channel.
Volume control: The Pump-Leak Model
cell
volume
[Na]i
pump rate
 RT N 
d
 (qwNi )  gNa V 
ln e  3pq
dt
F N i 

 RT K 
d
 (qwKi )  gK V 
ln e  2 pq
dt
F K i 

 RT C 
d
 (qwCi )  gCl V 
ln e 
dt
F Ci 

3 Na+
2 K+
ClX-
Na+ is pumped out. K+ is pumped in. So cells have low [Na+] and high [K+]
inside. For now we ignore Ca2+(for neurons only!). Cl- equilibrates
passively.
Charge and osmotic balance
 1
qw(N i  K i  Ci )  zx qX  qwe (N e  K e  Ce )
charge balance

X
N i  K i  Ci   N e  K e  Ce
w
osmotic balance
• The proteins (X) are negatively charged, with valence zx.
• Both inside and outside are electrically neutral.
• The same number of ions on each side.
• 5 equations, 5 unknowns (internal ionic concentrations, voltage, and
volume). Solve and analyze.
Steady-state solution
If the pump stops, the cell bursts, as expected.
The minimal volume gives approximately the correct membrane
potential.
In a more complicated model, one would have to consider time
dependence also. And the real story is far more complicated.
RVD and RVI
Okada et al.,
J. Physiol. 532, 3, (2001)
RVD and RVI
Okada et al.,
J. Physiol. 532, 3, (2001)
Lots of interesting unsolved problems
• How do organsims adjust to dramatic environmental changes (T.
Californicus)?
• How do plants (especially in arid regions) prevent dehydration in high
salt environments? (They make proline.)
• How do fish (salmon) deal with both fresh and salt water?
• What happens to a cell and its environment when there is ischemia?
Ion transport
• How can epithelial cells transport ions
(and water) while maintaining a constant
cell volume?
• Spatial separation of the leaks and the
pumps is one option.
• But intricate control mechanisms are
needed also.
• A fertile field for modelling. (Eg.
A.Weinstein, Bull. Math. Biol. 54, 537,
1992.)
The KJU model.
Koefoed-Johnsen and
Ussing (1958).
Steady state equations
N i  N m ev 
PNa v
 3qpNi  0
v
 1 e

K i  K sev 
PK v
 2qpNi  0
v
 1 e

Ci  Cse v 
PCl v
 0
v
1
e


w(N i  K i  Ci )  zX  0
N i  K i  Ci 

X
 N s  K s  Cs
w
Note the different current and pump
models
electroneutrality
osmotic balance
Transport control
Simple manipulations show that a solution exists if
3pq
Nm
PNa

N s 1 2 pq
PK
1
Clearly, in order to handle the greatest range of mucosal to serosal
concentrations, one would want to have the Na+ permeability a decreasing
function of the
mucosal concentration, and the K+ permeability an increasing
function of the mucosal Na+ concentration.
As it happens, cells do both these things. For instance, as the cell swells (due
to higher internal Na+ concentration), stretch-activated K+ channels open, thus
increasing the K+ conductance.
Inner medullary collecting duct cells
IMCD cells
Real men deal with real
cells, of course.
Note the large Na+ flux from
left to right.
A. Weinstein, Am. J. Physiol. 274
(Renal Physiol. 43): F841–F855,
1998.
Active modulation of the
membrane potential:
electrically excitable cells
Hodgkin, Huxley, and
the Giant Squid Axon
Hodgkin
Huxley
Don't believe people that
tell you that HH worked on a
Giant squid axon
The
reality
It was a squid
giant axon!

Resting potential
• No ions are at equilibrium, so there are continual background currents. At
steady-state, the net current is zero, not the individual currents.
• The pumps must work continually to maintain these concentration differences
and the cell integrity.
• The resting membrane potential depends on the model used for the ionic
currents.
gNa (V  VNa )  gK (V  VK )  0  Vsteady 
gNaVNa  gKVK
gNa  gK
linear current model
(long channel limit)
VF 
+
+
2  

F 2  [Na+ ]i  [Na+ ]e exp(VF

)
[K
]

[K
]
exp(
)
F
i
e
RT
RT
PNa  V 

P
V


 0

K 
VF
VF
1 exp( RT )
1 exp( RT )
RT  
RT  


RT PNa [Na+ ]e  PK [K + ]e 
 Vsteady 
ln

GHK current model
F  PNa [Na+ ]i  PK [K + ]i 
(short channel
limit)
Simplifications
• In some cells (electrically excitable cells), the membrane potential is
a far more complicated beast.
• To simplify modelling of these types of cells, it is simplest to assume
that the internal and external ionic concentrations are constant.
• Justification: First, small currents give large voltage deflections, and
thus only small numbers of ions cross the membrane. Second, the
pumps work continuously to maintain steady concentrations inside the
cell.
• So, in these simpler models the pump rate never appears explicitly,
and all ionic concentrations are treated as known and fixed.
Steady-state vs instantaneous I-V
curves
• The I-V curves of the previous slide applied to a single open
channel
• But in a population of channels, the total current is a function of the
single-channel current, and the number of open channels.
• When V changes, both the single-channel current changes, as well
as the proportion of open channels. But the first change happens
almost instantaneously, while the second change is a lot slower.
I  g(V,t) (V )
Number of
open channels

I-V curve of single
open channel
Example: Na+ and K+ channels
K+ channel gating
S00
S01
S0
S10
dx0
 bx1  2ax 0
dt
dx2
 ax1  2bx 2
dt
x 0  x1  x 2  1
2a
b
S1
a
2b
S11
x 0  (1 n) 2
x1  2n(1 n)
x2  n 2
dn
 a (1 n)  bn
dt
S2
Na+ channel gating
S00

2a
b

S10
S01

2a
b
a
2b

S11
S02

a
2b
Si
j

S12
x 21  m 2 h
dm
 a (1 m)  bm
dt
dh
  (1 h)  h
dt
inactivation
activation
inactivation
activation
Experimental data: K+ conductance
If voltage is stepped up and held fixed, gK
increases to a new steady level.
four subunits
gK  gK n 4
dn
 a (V )(1 n)  b (V )n
dt
 n (V )
rate of rise
gives n

dn
 n (V )  n
dt
time
constant
steady-state
Now fit to the data
steady state
gives n∞

Experimental data: Na+ conductance
If voltage is stepped up and held fixed, gNa
increases and then decreases.
gNa  gNa m 3 h
Four subunits.
Three switch on.
One switches off.
dh
 h (V )  h
dt
dm
 m (V )
 m (V )  m
dt
 h (V )
time
constant
steady-state
Fit to the data is a little more complicated now, but the same in principle.
Hodgkin-Huxley equations
applied current
dV
 gK n 4 (V  VK )  gNa m 3 h(V  VNa )  gL (V  VL )  Iapp  0
dt
dn
 n (V )  n (V )  n
generic leak
dt
dm
dh
 m (V )
 m (V )  m,
 h (V )  h (V )  h
dt
dt
C

•
•
activation
(increases with V)
much smaller than
the others
•
inactivation
(decreases with V)
An action potential
• gNa increases quickly, but then
inactivation kicks in and it decreases
again.
• gK increases more slowly, and only
decreases once the voltage has
decreased.
• The Na+ current is autocatalytic. An
increase in V increases m, which
increases the Na+ current, which increases
V, etc.
• Hence, the threshold for action potential
initiation is where the inward Na+ current
exactly balances the outward K+ current.
The fast phase plane: I
dV
 gK n 04 (V  VK )  gNa m 3 h0 (V  VNa )  gL (V  VL )  Iapp  0
dt
dm
 m (V )
 m (V )  m
dt
C

n and h are slow, and
so stay approximately
at their steady states
while V and m change
quickly
The fast phase plane: II
h0 decreasing
n0 increasing
As n and h change slowly, the dV/dt
nullcline moves up, ve and vs merge
in a saddle-node bifurcation, and
disappear.
Vs is the only remaining steady-state,
and so V returns to rest.

The fast-slow phase plane
Take a different cross-section of the 4-d system, by setting m=m∞(v),
and using the useful fact that n + h = 0.8 (approximately). Why? Who
knows. It just is. Thus
dV
 gK n 4 (V  VK )  gNa m3 (0.8  n)(V  VNa )  gL (V  VL )  Iapp  0
dt
dn
 n (V )  n (V )  n
dt
C
Oscillations
When a current is applied across the cell membrane, the HH
equations can exhibit oscillatory action potentials.
C
dV
 Iionic  Iapplied  0
dt
V
HB
HB
Iapplied
Where does it go from here?
•
Simplified models - FHN, Morris Lecar, Mitchell-SchafferKarma…
•
More detailed models - Noble, Beeler-Reuter, Luo-Rudy, … .
•
Forced oscillations of single cells - APD alternans, Wenckebach
patterns.
•
Other simplified models - Integrate and Fire, Poincare oscillator
•
Networks and spatial coupling (neuroscience, cardiology, …)