Download Worksheet: Understanding Membrane Potential

BBIO 351: Principles of Anatomy & Physiology I
Winter 2016
Worksheet: Understanding Membrane Potential
Goals
 Review fundamental information about cell membranes, ions, and electrochemical
gradients.
 Understand and apply the Nernst and Goldman-Hodgkin-Katz (GHK) equations.
 Determine whether an ion will go into or out of a cell if given the ion’s concentration
gradient and the cell’s membrane potential.
Group roles to assign
 Discussion leader: introduces questions, allots time for solo work, gathers input
 Lifeline: looks things up, gets instructor’s attention
 Equity officer: ensures equal participation
 Digression manager: keeps discussion on track
General background
Today we begin our coverage of the nervous system! Fundamentally, the nervous system is an
electrical system, i.e., messages are transmitted via the movement of charged particles (ions). As
you know from courses like BBio 220, the movement of a cation (such as Na+) into a neuron has
consequences very different from the movement of a cation out of that neuron. Since such ion
movements underlie the functioning of the entire nervous system, it is vital to understand exactly
why ions move in the directions that they move.
I. Two sides of the membrane
1. In Figure 1, place a large plus sign on the side of the membrane with the most positive charges
and a large minus sign on the side of the membrane with the fewest positive charges.
Figure 1: A polarized membrane. From Patrick J.P. Brown, Anatomy & Physiology: A Guided Inquiry
(2016).
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BBIO 351: Principles of Anatomy & Physiology I
Winter 2016
2. A membrane potential can be defined simply as the difference in electrical charge between
the two sides of a membrane (inside vs. outside). Based on #1 above, is the inside of a typical
cell more negative or more positive than the outside?
3. Figure 1 does not show any anions (negatively charged ions) on either side of the membrane.
Name at least one anion that contributes to the overall charge on one or both sides.
II. Electrical and chemical gradients
4. Does the membrane potential in #2 above – that is, the charge difference between the inside
and outside of the cell – pull K+ ions into the cell, or push them out of the cell? (This assumes
that K+ has open channels through which it can go in or out.)
5. Now consider potassium’s concentration gradient, as depicted in Figure 1. Does this
concentration gradient, in and of itself, attract K+ ions into the cell, or push them out of the cell?
Notice that your answers to #4 and #5 are opposites; that is, the electrical gradient (membrane
potential) favors movement of K+ in one direction, while the concentration gradient (chemical
gradient) favors movement of K+ in the opposite direction. We have arrived at the fundamental
concept that ions are governed by both gradients … and the fundamental dilemma that an ion’s
direction of movement can be tricky to predict when the two gradients oppose each other.
III. The Nernst equation: balancing electrical and chemical gradients
To figure out which gradient will “win” in a given situation, we have the Nernst equation. For
a given ion, the Nernst equation tells us the membrane potential (that is, the electrical gradient)
that perfectly counterbalances that ion’s chemical gradient, so that there is no net movement of
the ion into or out of the cell. This special membrane potential is called that ion’s equilibrium
potential (abbreviated E for equilibrium) or Nernst potential.
The Nernst equation can be written in several forms, depending on one’s assumptions. Figure 2
shows a “full” version and a simplified version, along with (of course!) the lyrics for a Nernst
equation jingle.
Note the logarithm in the equation. A logarithm is another word for exponent. For example,
1000 may be written as 103, so log10(1000) is 3.
6. If the ratio [ion]extracellular/[ion]intracellular is less than 1, the log10 of this ratio will be _________.
Therefore, any cation that is more concentrated inside the cell than outside (e.g., K+) will have an
equilibrium potential (E) that is _______.
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BBIO 351: Principles of Anatomy & Physiology I
Winter 2016
Figure 2: Nernst equation and song lyrics. From https://www.youtube.com/watch?v=XfxwK9mTlkw.
7. If the ratio [ion]extracellular/[ion]intracellular is greater than 1, the log10 of this ratio will be _______.
Therefore, any cation that is more concentrated outside the cell than inside (e.g., Na+) will have
an equilibrium potential (E) that is _______.
8. Chloride (Cl-), like Na+, is more concentrated outside the cell than inside, but its valence (z) is
negative (-1, to be exact). What can you conclude about chloride’s equilibrium potential (ECl)?
9. Use the Nernst equation to find ECl for a cell whose extracellular [Cl-] is 110 mM and whose
intracellular [Cl-] is 5 mM. You may use a calculator. Does your answer match your conclusion
in #8 above?
IV. A graphical approach to electrochemical gradients
The Nernst equation tells you the membrane potential at which the electrical and chemical
gradients are exactly counterbalanced. Generally, though, we will want to know whether a
specific ion will flow in or out at a specific membrane potential that is not the equilibrium
potential.
Here is Dr. C’s recommended method for determining the direction of an ion’s flow at any
membrane potential:
A. Find the ion’s equilibrium potential (E).
B. Set up a graph with membrane potential on the X axis and overall driving force on the Y
axis.
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BBIO 351: Principles of Anatomy & Physiology I
Winter 2016
C. Plot 2 easy points: the X-intercept (when Y=0) and the Y-intercept (when X=0).
D. Connect the dots!
The chemical gradient is assumed to be constant throughout this process.
To see how this method actually works, let’s do an example with Cl- ions, using the information
given above.
Step A: See #9 above. (Did you get -78 mV?)
Step B. Note that
“net driving force”
(Y axis) represents
the combined
influence of the
electrical and
chemical gradients.
Step C. The Xintercept, when
Y=0, is simply the
ECl you calculated
from the Nernst
equation. The Yintercept is the
point representing
the net driving
force when there is
no electrical
gradient, i.e., when
the membrane
potential is 0 mV.
In this case, there is
only a chemical
gradient, and since
you know that Cl- is
more concentrated
outside the cell, that
gradient drives Clinward.
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BBIO 351: Principles of Anatomy & Physiology I
Winter 2016
Step D. Once you
draw a line through
the 2 points, this
line will show you
the direction of the
ion’s flow (in or
out) at any
membrane
potential. Notice
that all possible
membrane
potentials can be
divided into the 3
regions labeled at
the bottom.
10. Based on the method and data above, which way will Cl- flow (into the cell or out of the cell)
at a membrane potential of +78 mV?
11. Based on the method and data above, which way will Cl- flow (into the cell or out of the cell)
at a membrane potential of -90 mV?
12. Now try the following problem, which was on a quiz in a previous quarter….
Imagine an alien animal with neurons like ours except with different ions, different ion channels, and a
resting membrane potential of -100 mV. The Nernst equation still holds true.
If ion X2+ is at a concentration of 10 mM inside the cell and 100 mM outside the cell, fill in each empty
box of the following chart with INTO CELL, OUT OF CELL, or NEITHER. You may use IN/OUT/NEITHER for
short. Note that log10(1/10) = -1 and log10(10) = 1.
Direction X2+ is driven,
Direction X2+ is driven,
Direction X2+ is driven,
Membrane considering only the
considering only the
considering the overall
potential
electrical gradient
chemical gradient
electrochemical gradient
-100 mV
-58 mV
-29 mV
0 mV
29 mV
58 mV
100 mV
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BBIO 351: Principles of Anatomy & Physiology I
Winter 2016
V. The Goldman-Hodgkin-Katz (GHK) equation: calculating membrane potential based on
multiple ions
The Nernst equation only considers one ion at a time. The Goldman-Hodgkin-Katz equation
essentially combines the Nernst equations for multiple ions to calculate a membrane potential
(Vm; V is for voltage) based on these ions’ intracellular and extracellular concentrations and the
membrane’s permeability to these ions. Below is one form of the GHK equation (with subscripts
i for inside and o for outside).
Figure 3: GHK equation and song lyrics. From https://www.youtube.com/watch?v=jhoREcHVCj8.
Permeability reflects the density of open ion channels; the more open channels there are, the
higher the permeability. The membranes of resting neurons are about 25 times more permeable
to K+ than to Na+ because more K+ channels are open.
VI. Computer modeling of neurons
We will now use a computer program to further explore the Nernst and GHK equations.
Go to http://nernstgoldman.physiology.arizona.edu and click the “Launch now” button (or
download the stand-alone program). Notice the four tabs at the upper right: Nernst, Nernst
@37°C, Goldman, and Goldman @37°C. Also notice the lower-right menu of ion/permeability
presets: default, generic cell, squid axon, skeletal muscle, red cell.
Load the preset of a generic (mammalian) cell.
13. According to this program, what are the extracellular and intracellular Na+ concentrations
[Na+]o and [Na+]i (o is for outside and i is for inside)?
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BBIO 351: Principles of Anatomy & Physiology I
Winter 2016
14. What is ENa for this generic cell, according to this program? (Is this a Nernst question or a
GHK question?) Confirm that the sign of your answer (positive or negative) is what you would
expect, based on the ion’s charge and concentration gradient.
15. Much of the pioneering work on how neurons function was done with squid, a marine animal
whose bodily fluids somewhat resemble seawater. What are the extracellular and intracellular
Na+ concentrations for a squid axon, according to this program?
16. How does a squid axon’s ENa compare to a generic mammalian cell’s ENa?
17. Now use a Goldman tab to determine the membrane potential (Vm) of a generic mammalian
cell and a squid axon. How do the two compare?
18. Based on your above answers, does evolution seem to keep any or all of the following
parameters within narrow limits?
 extracellular ion concentrations
 intracellular ion concentrations
 equilibrium potentials
 membrane potentials
19. Notice that, in the Goldman tabs, the relative permeabilities (PK and PNa) are adjustable.
Increase the membrane’s permeability to Na+. How does the membrane potential change? To
what phase of an action potential does this change correspond?
20. Now bring the Na+ permeability back to its original value and increase the K+ permeability.
What happens to the membrane potential now?
22. According to the GHK equation, the membrane potential (Vm) will always be somewhere
between ENa and EK. What determines whether Vm is closer to ENa or closer to EK?
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