Membranes and Transport
Topic II-1
Biophysics
Nernst Equation
k BT
Co  RT

 Co 
V 
 ln 


ln

 C
C
i
i


q
F
www.kcl.ac.uk/teares/gktvc/vc/lt/nol/Nernst.htm
Co
At 300 K get V  60 mV log
Ci
• F = 96,500 Coulomb/mole
•Simplest equation for membrane potential – one ion
Goldman-Hodgkin-Katz Equation
 
 
 
 
  Pi C  i 1   Pi C  i 

k T 
i
2
V  B ln  i  



q
P
C

P
C
i
i
i i
2
 i

1
 i
 
 
 
 
 
 
k BT  Pk K  o  PNa Na  o  PCl Cl i 

V 
ln 



q
P
K

P
Na

P
Cl
i
i
o 
Na
Cl
 k
• There is one membrane potential that counters all concentration
gradients for permeable ions
• Pi = (mikBT)/d = Di/d, with D – diffusion constant [cm2/sec]
Example – simple Neuron
    
    
 K  o  b Na 
V  58 mV log  

K

b
Na
i

o
with b = PNa/PK
i
b = 0.02 for many neurons (at rest).
[K]i = 125 mM
[K]o = 5 mM
[Na]i = 12 mM
[Na]o = 120 mM
What is V?
• Note: Can define Nernst potential for each ion, Vk = -80
mV; VNa = +58 mV. Relative permeabilities make
membrane potential closer to Vk
Do Soma 1 (Nernst)
Soma 1 Nernst Potential
Soma 1 – Soma Nernst
Temperature is 20 C. This model is for a membrane that is only permeable to potassium. Top
plot is VK vs time. Bottomt plot is VK vs log([K]out)
1. Run (play). What should VK be? [Note R = 8.31 J/mol*kelvin, F = 96,500 colomb/mole]
2. Double K out, -> 10.2
What happens to VK?
3. When is VK zero?
4. Is VK linearly dependent on log(K]out)? Try different Kout: 100, 0.1.
• Alt enter gives full screen
• Use left and right mouse clicks to change Ckout (concentration of
potassium outside cell)
Electrical Model
Vm
gV


g
i
i
i

g Cl VCl  g K VK  g Na VNa  g Ca VCa
g Cl  g K  g Na  g Ca
(from Kirchoff’s laws)
g is like conductance (=1/R) and like permeability
This equation is equivalent to Godman-HodgkinKatz equation.
Example squid axon
IK = gK (Vm – VK), Vm = -60 mV
IK = gK (-60 – (-75)) mV = gK(+15 mV).
[K]in =125mM
g always positive.
V = Vin – Vout
Vm = -60
VK = -75
positive current = positive ions flowing
out of the cell.
[K]out = 5 mM
Vm not sufficient to hold off K flow so
ions flow out. When Vm = Vk then no
flow.
• Remember electric field
points in direction of force on
positive test charge
• E always points from higher
potential (for ions)
Vions  Vin  Vout  
in
[K]in =125 mM
E
ds
[K]out = 5 mM
E
 
E  ds; Nernst potential opposes this
out
Do Soma 3
(Resting
Potential)
Soma 3 Resting Potential
Soma 3 – Soma – resting
Three ions: Na, K, Cl. Top graph is VK, VNa, VCl, Vm (E used instead of V) vs time.
Bottom graph is IK, INa, ICl, Im vs time
1. Run (play). Why is ICl so low? What is total current?
2. Set gNa=gCl=0. [These are written as QNa etc] You can keep the simulation going as
you do this. What happens to Vm?
3. Set gCl = 5, leave gK as 16, Set gNa really high = 100 etc. What happens to Vm?. This
is like action potential.
Vm
gV


g
i
i
i

g Cl VCl  g K VK  g Na VNa  g Ca VCa
g Cl  g K  g Na  g Ca
Donnan Rule and other considerations
Example of two permeable ions and one
impermeable one inside
•
•
•
•
•
KoClo = KiCli Donnan Rule
Electroneutrality
Osmolarity
Goldman- Hodgkin-Katz
Apply electroneutrality inside and out and
plug in Donnan rule and get


 [K]o 
2
K
o

  58 mV log 
Vm  58 mV log 
2
2 2 1/ 2


[K]


4
K

A
Z

AZ
i 

o


Animal Cell Model
Ci (mM)*
Co (mM)
P>0?
K+
125
5
Y
Na+
12
120
N**
Cl-
5
125
Y
A-
108
0
N
H2O
55,000
55,000
Y
* Should really use Molality (moles solute/ kg solvent) instead
of per liter – accounts for how molecules displace water (nonideality).
** More on this later
Maintenance of Cell Volume
Cell
impermeable to
sucrose
http://www.himalayancrystalsalt.com/html/images/PAGE-osmosis.gif
www.lib.mcg.edu/.../section1/1ch2/s1ch2_25.ht
• Osmolarity must be same inside and out
• Concentration of permeable solutes must be
same inside and out
• Si = So and Si + Pi = So (Osmolarity)
• Solutions: cell wall, Pwater = 0, Pextra cellular solutes = 0
Animal Na impermeable model
• Apply electroneutrality outside, Donnan, and
osmolarity
• Get unknowns and Vm = -81 mV
Active Transport
 
  
    
 K    Na  

 V ln 
 K    Na  
k BT  Pk K  o  PNa Na 
Vm 
ln 


q
P
K

P
Na
i
Na
 k
 Vm

i

o
o
o

o

i
i
with  = (n/m)(PNa/PK),n/m = 2/3  Vm  VK.
http://faculty.ccbcmd.edu/~gkaiser/biotut
orials/eustruct/sp1.html
•
Na-K Pump
-
Two sets of two membrane spanning subunits
Phosphorylation by ATP induces a conformational change in the protein allowing pumping
Each conformation has different ion affinities. Binding of ion triggers phosphorylation.
Shift of a couple of angstroms shifts affinity.
Exhibits enzymatic behavior such as saturation.
Electrical Model
Vm 
g Cl VCl  g K VK  g Na VNa  g Ca VCa  I p  I
g Cl  g K  g Na  g Ca
Now include current for pump, Ip as well as input current I.
Do Soma conductance and Na pump)
Soma 4 Conductance
Goal is to study how to get membrane conductance and that for each ion and Nernst potentials.
Have Na, K, and Cl ions. Units of conductance are nanosiemens and current is in nanoamps.
Have Vm vs time and vs input current.
1. Run. Get Vm = -63.
2. Set g for Na and Cl = 0. Run for different I stim amplitude (-4 to 4) How get gK?
[Hint: Have Vm = (gkVK + I)/gK or Vm = (1/gK)I + VK]
How get Vk?
3. HW will be to repeat for Na.
Vm 
g Cl VCl  g K VK  g Na VNa  g Ca VCa  I p  I
g Cl  g K  g Na  g Ca
Soma 5 Na Pump
Look at contribution of Na pump contribution to Vm and role of intracellular Na ions
is setting the pump rate. Have Vm vs time and INa, [Na]in, and INapump vs time.
Note that in equilibrium, current of Na pump and Na equal and opposite.
1.
Run. Pump off (Na NaPumpmax =0) Get Vm = -67 mV. [Na]in = 10 mM.
2.
Set NaPumpmax to 60 (fM/s). Run for a bit. [Na]in still about 10 mM but
Vm now about -74 mV.
3. Increase max pump rate 145, 245 . [Na]in decreases and hyperpolarization is
reversed. Why?
4. Put NaPumpmax back to 60. InjectNaStimon is on and have amplitude at nA.
Fire. Now depolarization is great and [Na] inside goes up (ofcourse ) and then
down.
Patch Clamping
www.essen-instruments.com/Images/figure2.gif
• Invented by Sakmann and Neher [Pflugers Arch 375: 219-228, 1978]
•Can be used for whole cell clamp (measure currents in whole cell, placing
electrode in cell) like on left or pulled patch as on right (potentially measure single
channel).
•Can control [ions].
•Usually voltage clamp (command voltage or holding voltage) and observe current
(I = gV). Ix = g(Vh-Vx) where x is for each ion and Vx is Nernst potential for that ion. With
equal concentration of permeable ion on both sides, get g easily
Voltage Gated Channels
• There is a degree of randomness in opening and
closing of channels
•Proportion of time open is proportional to Voltage
for some channels
•Average of many channels is predictable
• When [ions] not limiting, can get nernst potential
when current reverses
I = gx*(Vh – Vx)
Multiple Channels
I
t
• Get several channels on a patch
– Gives quantized currents
• Parallel: geq = S gi; Series: 1/geq = S 1/gi
– g = 1/R
Ligand gated channels
Nicotine also binds to Ach receptor
– called nicotinic receptor
• When Ach binds, gate opens and lets in Na+
and K+.
• I is proportional to [Ach]2 (binds 2 Ach)
Do Patch Cl, K, Multiple K, ligated)
Patch 1 Cl
Single Cl channel. Plot is chloride current in pA vs time. Will have VCl =0 since
concentrations of ions on both sides set to zero.
1. We want to have VCl =0, so scroll down in the parameter window and set ECL
= 0.
2. Run. Calculate g. (Note Vhold is 50 mV). I = gV;
3. Change Vh. 75, 99, -50 etc. Is the current proportional to the holding voltage?
Is this channel voltage gated?
Patch 2 K
Same as Cl one, but now have K.
HW – calculate g. Show work and screen shot.
1.
Run.
2.
Change Vh. -60, -70,-40, -30. Is the
current proportional to the holding voltage? Is
this channel voltage gated?
Patch 4 Multiple K channels
Have VK = -80 mV due to unequal concentrations of K on each side of patch.
I = g (Vh – Vk)
1.
Hit play. KCHNumber = 1 (one channel). I = g(-50--80) so current is positive.
What is the average current? This will be current when it is open multiplied by fraction
of time open.
2. Increase KCHNumber to 2. What do you notice? Why are there sometimes twice as much
current?
3. Now increase KCHNumber to 3. What do you notice? Why are there only occasionally
three times as much current?
4. Increase KCHNumber to 100. [Use two arrows facing each other to chnge y-axis scale]
Why do we see what we see?
Patch 5 Ligand gated channels
Single ligand gated channel. Plot current through channel (ISynsum) vs time. Vhold = -60 mV lik
in a cell. Note here, play goes for one cycle.
1. Run. [Ach] = 0 (stimulator off). See brief random openings.
2. Hit fire [toggle Stim to on], which gives 5 units of Ach for 0.5 s repeatedly, and then
hit play.
3. Increase amplitude to 10 (increases Ach [stim amplitude]). Notice it is open more. It
is cooperative. It also saturates.
4. Verify that reversal potential is -10 mV. Set the number Ach channel to 10, Num
SynChannel and increase [Ach] (amplitude) to 10. Take Vhold from -60 mV to 20 mV.
(Use Y-axis zoom out if you need to).
I = g(Vh-VACH) = g*Vh- g*VACH-. Reversal potential is when I is zero. See get zero
current when Vh = -10 mV.
5. With the number of channels at 100, re-explore the dependence on [Ach] taking Ampl
from 1 on up to 40. Rescale the y-axis from -200 to 200 [take Vhold back to -60mv
first] Do you see linear dependence on concentration of Ach? Do you see saturation?
Facilitated Transport
Co Yo Ci Yi
K

CYo
CYi
 Co
Ci 
J  J max 


 K  Co K  Ci 
Jmax = NYDY/d2
NY = number of carriers
DY = diffusion constant of Carrier
d = membrane thickness
• Get Saturation Kinetics
• Lower activation energy
Jmax
Co