Problem set 1

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NS201A
Fall 2010
Problem set 1.
1. Derive using the N ernst-Planck flux equation an expression for the voltage
across a m em brane w hen net current = 0. Assum e only N a + , K+ , and Cl- are
perm eant, so Itotal = IN a + IK + ICl and I = F z iJi, w here F is the Farad ay constant
2. Tw o solutions are separated by m em brane of thickness = d. The m em brane is
perm eable only to K+ and Cl- and the fluxes are d escribed by the N ernst-Planck
flux equation assum ing constant field . Mem brane thickness d = 5 x 10 =7 cm w ith
d iffusion coefficients K+ = D K and Cl- = D Cl and partition coefficients K+ = k k+ and
Cl- = k Cl-. Find:
a. an expression of fluxes of K + and Cl- as a function of the voltage across the
m em brane.
b. an expression for current d ensity as a function of the transm em brane vo ltage
w here F z iJi
c. an expression for the voltage in the cond ition of zero net m em brane current.
What is the num erical value of the current if:
D K+ = D Cl- = 5 x 10-8 cm 2/ second
KK+ = 10-3 and k Cl- = 2 x 10-4
Concentration of KCl on sid e 1 = 10 -2 M and sid e 2 = 10-1 M
3. In certain pathological cond itions, such as d ehyd ration, renal failure, cardiac
glycosid e poising and seizure activity in the brain, large changes in the
extracellular K concentration occur. Changes in extracellular K concentration can
have large effects on m em brane electrical excitability, m ost dram atically in cells
w here the resting potential d epend s on the inw ard ly rectifying K channel. The
effect on the m em brane potential via K ir w ill differ in different cell types
d epend ing on the P N a / P K ratio.
The exam ple below gives values for N a and K ion concentrations and
perm eabilities in pacem aker versus non -pacem aker cells in the heart. Card iac
m uscle contains prim arily Kir. The values are given for norm al, low
(hypokalem ia), and high (hyperkalem ia) serum K concentrations. Calculate: a)
the equilibrium potential for N a and K using the N ernst equation; b) the resting
m em brane potential E m using the Gold m an-Hod gkin-Katz equation.
NS201A
Fall 2010
PN a
N ao N ai
PK
Ko
Ki
N on-p acem aker (N PM)
norm al K
1
150
15
200
5
120
N PM: hyperkalem ia
1
150
15
400
7.5
120
N PM: hypokalemia
1
150
15
100
2.5
120
Pacem aker (PM)
norm al K
1
150
15
20
5
120
PM: hyperkalem ia
1
150
15
40
7.5
120
PM: hypokalem ia
1
150
15
10
2.5
120
EN a
EK
Questions:
a) What is the effect of hyperkalem ia on the m em brane potential E m of nonpacem aker cells? Does it d iffer from the K equilibrium potential?
b) What is the effect of hypokalem ia on E m in non-pacem aker cells?
c) What is the effect of hyperkalem ia on Em in pacem aker cells? Does it differ
from non-pacem akers? Why? What effect w ould this have on pacem aker
frequency (heart rate)
d ) What is the effect of hypokalem ia on E m in pacem aker cells? Does it d iffer
from non-pacem akers? Why? Does hypokalem ia increase or d ecrease heart
rate?
4. Suppose a neuron contains only K + as the internal cation at a concentration of
[K+ ]i and N a + as the only external cation at a concentration [N a + ]o. Design an
experim ent to d etermine the perm eability ratio P N a / P K of the sod ium channel
Em
NS201A
Fall 2010
using the constant field equation. What is the m eaning of the perm eability
constant.
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