Neurobiology W3004
Fall 2003
Problem Set #2: The Resting Membrane Potential
1. Imagine a beaker containing a semi-permeable membrane that is selective to
K+ ions only. The membrane separates solutions containing 100 mM KCl on the
left, and 1 mM KCl on the right.
A. What potential is developed across the membrane?
B. How long will that potential persist?
C. If the KCl concentration on side 2 is increased successively by 2, 5, 10, 20
and 100 fold, what will be the new equilibrium potentials? What does a graph of
concentration vs. potential look like? Is the potential related to the concentration
change in a linear or non-linear fashion?
D. If 100 mM NaCl were added to the left side of the beaker, what would be the
effect on the potential difference between the two sides?
2. The major external and internal ion concentrations for the giant cell R15 of
the marine mollusk Aplysia californicus are approximately:
[K]o = 10 mM
[K]i = 280 mM
[Na]o = 485 mM
[Na]i = 61 mM
[Cl]o = 485 mM
[Cl]i = 51 mM
A. What are the equilibrium potentials for each of these ions?
B. If, by the magical addition of electrical current, the cell membrane were forced
to a potential of 0 mV, how would each ion redistribute itself assuming that the
membrane is permeable to each ion. That is, at equilibrium, what would be the
new internal and external concentrations of each ion at a membrane potential of
0 mV?
3. For the cell of problem 2 the actual resting membrane permeabilities are:
PK : PNa : PCl = 1.0 : 0.12 : 1.44
A. What is the resting potential predicted by the GHK equation?
B. What would be the resting potential if external K+ were increased by 5 fold?
C. What would be the resting potential if external Na+ were decreased by 5 fold?
D. What would be the resting potential if PNa were increased 5 fold?
4. With a microelectrode you inject a 100 msec long pulse of 0.5 nA of current
into a cell and thereby drive the membrane potential of the cell from its resting
level of -50 mV to +50 mV. The capacitance of the cell membrane is 25 pF.
What is the membrane time constant? What is the membrane potential 15
msec after the end of the current pulse?
A particularly instructive problem:
How many ions must be transferred from one side of the membrane to the other
to change the membrane potential in a spherical cell of diameter 50 microns
(50x10-6 m) by 100 mV (1x10-1 V)?
You will need to know the following:
1. The relationship between the amount of charge separated (i.e. moved
from one side of the membrane to the other) and the membrane potential
is given by the electrical capacitance of the membrane. The lipid bilayer
membrane acts as a capacitor and has a capacitance value of 1
microfarad per square cm, or 1x10-6 F /cm2.
2. The relation between charge and capacitance is:
Q = CV
where Q is the charge is coulombs, C is the capacitance in farads, and V
is the potential in volts.
3. There are 96,500 coulombs in one mole of a univalent ion such as K+.
From these parameters you should be able to determine the number of K+ ions
that must be transferred across the membrane to change the membrane
potential by 100 mV.
If we assume that the cell has a normal internal concentration of 100 mM K+,
then what is the ratio of ions in flux to those already in the cell? That is, for every
K+ ion that must move across the membrane how many are already in the cell?
Is it likely that influx of K+ ions will significantly change the intracellular
concentration?
A Special Advanced Problem:
Why does the gas constant, R, appear in the Nernst equation? How can R be
related to Voltage?