1. Membranes and Transport
Biophysics

2. Nernst Equation Simplest equation for membrane potential – one ion
RT/F = volts; ln(x) = 2.3log(x)
6.022E23; E-19
Faraday constant
Z is charge of ion here
F = 96,500 Coulomb/mole
Simplest equation for membrane potential – one ion

3. Goldman-Hodgkin-Katz Equation
P for organic ions = 0
There is one membrane potential that counters all concentration gradients for permeable ions
Pi = (mikBT)/d = Di/d, with D – diffusion constant [cm2/sec]

4. Example – simple Neuron
with b = PNa/PK,
b = 0.02 for many neurons (at rest).
[K]i = 125 mM [K]o = 5 mM
[Na]i = 12 mM [Na]o = 120 mM
Example: What is DV? Compare to nernst potentials (verify values below).
Temperature = 20 degrees C
298 K
Note: Can define Nernst potential for each ion, DVk = -81 mV; DVNa = +58 mV. Relative permeabilities make membrane potential closer to DVk
Do Soma 1 (Nernst)

5. Soma 1 Nernst Potential
Alt enter gives full screen (and goes back to partial screen)
Use left and right mouse clicks to change Ckout (concentration of potassium outside cell)

6. Basics of Electrical Model
V = IR = I/g. g = 1/R
g is like conductance, [g] = mho or siemens
If have two batteries add V
Eg INa = gNa(Vm-VNa)
When Vm=VNa, no current.

7. Electrical Model g is like conductance (=1/R) and like permeability
(from Kirchoff’s laws)
g is like conductance (=1/R) and like permeability
This equation is equivalent to Godman-Hodgkin-Katz equation.

8. Example squid axon
IK = gK (Vm – VK), Vm = -60 mV
IK = gK (-60 – (-75)) mV = gK(+15 mV).
g always positive.
DV = Vin – Vout
positive current = positive ions flowing out of the cell.
Vm not sufficient to hold off K flow so ions flow out. When Vm = Vk then no flow.
[K]in =125mM
[K]out = 5 mM
Vm = -60
VK = -75
Arrows next to V in top diagram just indicating counterforce against concentration gradient.
Going from high to low potential.
DeltaV = Vfinal-vinitial
Remember electric field points in direction of force on positive test charge
E always points from higher potential (for ions)
[K]in =125 mM
[K]out = 5 mM E ds
Do Soma 3 (Resting Potential)

9. 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
Run (play). Why is ICl so low? What is total current?
Set gNa=gCl=0. [These are written as QNa etc] You can keep the simulation going as you do this. What happens to Vm?
Set gCl = 5, leave gK as 15, Set gNa really high = 100 etc. What happens to Vm?. This is like action potential.

10. 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
2016-skip slides 10-14

12. Animal Cell Model
Ci (mM)*
Co (mM)
P>0? K+
125 5 Y
Na+ 12
120
N**
Cl- A-
108 N
H2O
55,000
* Should really use Molality (moles solute/ kg solvent) instead of per liter – accounts for how molecules displace water (non-ideality).
** More on this later

13. Maintenance of Cell Volume
Cell impermeable to sucrose
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

14. Animal Na impermeable model
Apply electroneutrality outside, Donnan, and osmolarity
Get unknowns and Vm = -81 mV

15. Active Transport with a = (n/m)(PNa/PK),n/m = 2/3  Vm  VK. Na-K Pump
Pump sodium out
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.

16. Electrical Model
Do soma 4 and 5
Now include current for pump, Ip as well as input current I.
Do Soma conductance and Na pump)

17. Soma 4 Conductance

18. 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.
Increase max pump rate 145, [Na]in decreases and hyperpolarization is reversed. Why?
Put NaPumpmax back to 60. InjectNaStimon is on and have amplitude at 10 nA. Now depolarization is great and [Na] inside goes up (ofcourse ) and then down.

19. Patch Clamping
Invented by Sakmann and Neher [Pflugers Arch 375: , 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

20. 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
PATCH EXERCISES
When [ions] not limiting, can get nernst potential when current reverses
I = gx*(Vh – Vx)

21. Multiple Channels Get several channels on a patch
Gives quantized currents
Parallel: geq = S gi; Series: 1/geq = S 1/gi
g = 1/R

22. Example
You are studying a single potassium channel with VK = - 80 mV clamped at – 50 mV. The current oscillates between 0 and 12 pA. Out of 1000 seconds it is open for 150 s.
What is the average current?
If you had 100 such channels in parallel, what would the current be?

23. 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 1,2,4,5

24. 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.
We want to have VCl =0, so scroll down in the parameter window and set ECL = 0.
Run. Calculate g. (Note Vhold is 50 mV). I = gV;
Change Vh. 75, 99, -50 etc. Is the current proportional to the holding voltage? Is this channel voltage gated?

25. Patch 2 K HW – calculate g. Show work and screen shot.
Same as Cl one, but now have K.
HW – calculate g. Show work and screen shot.
1. Run.
2. Change Vh. -60, -70,-40, Is the current proportional to the holding voltage? Is this channel voltage gated?

26. Patch 4 Multiple K channels