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Resting Membrane Potential 101011


									                                                         Header to be Filled in by Amy Gelban


                                        Tom Shannon
                                 AIM, Google, Yahoo: tshanno

Resource Material: (Recommended reading assignments)
Reading: Berne and Levy Principals of Physiology, 4th Edition, p. 21-29

Lecture objectives:
1. Understand the forces involved in the movement of ions in solution and across membranes
2. Understand the definition of transmembrane potential (i.e., the difference between
    intracellular and extracellular potentials); contrast this with extracellular measurements such
    as the ECG.
3. Understand chemical and electrical forces qualitatively.
4. Understand the processes that are involved in a permeable ion coming into electrochemical
    equilibrium across a cell membrane.
5. Memorize the Nernst equation, and be able to apply it to important ions (e.g., K+, Na+, Cl-,
6. Understand the concept of electroneutrality; know the degree to which it is an approximation
    and how good of an approximation it usually is.
7. Approximately know the typical concentrations of Na+, K+, Cl- and Ca2+ inside and outside
    the cell; also know the approximate equilibrium potentials for these ions in typical cells (see
    Table 2.1).
8. Understand what is meant by membrane capacitance, and know the basic factors (e.g.,
    membrane area and thickness – or „effective‟ membrane thickness) that influence cell
9. Learn about the function and properties of the transverse tubular system of skeletal muscle
    and of cardiac muscle.
10. Understand the basic „take home messages‟ of the GHK and chord conductance equation,
    e.g., that ionic species with the highest membrane permeability (or conductance) have the
    greatest effect on the membrane potential.
11. Recognize that an ion to which the membrane is permeable but which is not actively
    transported will, in the steady state, have an equilibrium potential equal to the membrane
    potential (resting potential). Cl- is an example of this in many cells.
12. Understand the consequences of hyperkalemia in terms of the membrane resting potential.

Key Words:
anion          equilibrium potential
capacitance    Goldman-Hodgkin-Katz (GHK)
cation         Nernst Equation
electrogenic   permeability
I. Diffusion of Ions in Solution and across Membranes: In the molecular and
   cellular block lecture, we focused primarily on the movement of uncharged solutes both in
   solution and across membranes (although many of the molecules involved in protein-
   mediated transport move charged molecules). This is vital, but it is a small part of the story
   of solute movement. Most of the solutes (typically about 80-90%) in both the intracellular
   and extracellular spaces are charged. Charged solutes are called ions. Positively charged
   ions are called cations, while negatively charged ions are called anions. When ions diffuse,
   they also obey Fick‟s Law, but electrical forces must also be taken into account.

   1. When we consider the movement of ions, in addition to uncharged solutes, across
      membranes other rules must also be considered. Some rules have already been
      described, but will be briefly repeated here; others are new:

       a.   All permeable uncharged solutes will eventually (in steady state) have the same
            concentration on both sides of the membrane (unless they are actively transported).
            As you will see, this is not true of charged solutes even if they are not actively
            transported (consider chloride as described below).

       b.   All impermeant solutes will (taken as a group) have the same concentration on both
            sides of the membrane. This does NOT apply to individual solute species – e.g.,
            Na+ or plasma proteins, or other solutes that are either impermeable, or maintained
            effectively impermeable due to the action of pumps or secondary active
            transporters. What IS required is that the total concentration of ALL impermeant
            (or effectively impermeant) solutes taken together be the same on both sides of the

       c.   Electroneutrality must be maintained on both sides of the membrane. This is a
            reasonable approximation to reality. If a membrane potential is present (as is the
            case across essentially all cell membranes) then there will be a small violation of
            electroneutrality. This is obviously necessary, since if one side of the membrane
            has a positive potential relative to the other side it must also have a positive
            charge. But the magnitude of this charge is normally extremely small relative to
            the bulk concentration of ions in the intracellular and extracellular fluids. For
            example, as will be described below, creating a resting membrane potential
            difference of about 60 mV across the cell membrane typically only requires that the
            imbalance of + and – charges on the two sides of the membrane be about 0.0001 to
            0.001%. This is not detectable by normal chemical assays, and can therefore be

       d.   These are all the rules that are needed to determine cell concentrations and volumes
            in PASSIVE situations (i.e., situations in which pumps and other forms of active
            transport are not involved). Unfortunately, such situations are rare, if, in fact, they
            ever occur in normal physiological situations. What is not rare is that these rules
            make an adequate approximation for many short-term clinical applications. These
            are the sorts of approximations that make such issues manageable. You should be
(Recall from Cell and Molecular that you are NOT responsible for the
material in commentaries. They are there to supplement the notes and to
help your understanding.)

What is Voltage?

Voltage is usually represented in equations by the symbol V or E. Many
academics prefer the use of E because V is also the standard symbol for
Volt, the unit which quantitates voltage. We will use both abbreviations
interchangeably in this class.

Voltage is a measure of the amount of work it takes to separate two groups
of charges. Normal solution is electroneutral and it takes energy to
separate charges in solution. In physiology, this separation is normally
maintained by a membrane which stretches between groups of charged ions.
If the membrane is made slightly permeable to one or more of these ions,
they will flow across the membrane in an effort to join their opposites on
the other side. This flow is measured as electrical current and it is
proportional to the voltage.

When we talk about voltage in this class we will actually be talking about
the difference in voltage (V) between two points. That is, we talk about
the voltage at one point relative to a reference voltage at another point.
Physiologically this usually means the voltage inside the cell membrane
relative to the voltage outside the membrane (again, by convention). It is
also termed the “potential difference” Simply put, you can think of the
        aware of complications that a measure of how much ions break-down (e.g.,
membrane potential difference as arise when such approximations“want” to
        consequences of the the cell because pump for cell volume regulation).
move either into or out offailure of the Na+/K+of their charge.

2. The Resting Membrane Potential: The cell membrane resting potential is simply the
   difference in potential across the cell membrane that occurs when the cell is at rest –
   i.e., when it is not involved in transmembrane potential changes such as synaptic
   potentials or action potentials. You have already learned about transmembrane
   potentials, and about extracellular potentials frequently measured clinically such as the
   ECG. It is important to remember that transmembrane potentials can only be measured
   as the difference between the intracellular and extracellular potential (e.g., with a pair of
   glass microelectrodes, one of which penetrates the cell membrane and the other of
   which is placed just outside of the cell). Obviously such invasive measurements are
   only very rarely attempted in clinical situations. By tradition, the cell transmembrane
   potential (Vm) is measured as the intracellular potential (Vi) minus the extracellular
   potential (Ve), i.e.,
                                   Vm = Vi -Ve
    The resting potential of various cells within the human body ranges from as little as
    about –10 mV to as much as about –90 mV. Cardiac and skeletal muscle cells
    generally have resting potentials of about –90 mV. Nerve cells usually have somewhat
    smaller resting potentials in the neighborhood of –70 mV.

3. Because much of what follows may seem to be only distantly related to clinical
   medicine, I want to pause now to tell you where we are going and why we are going

    The fact that cells have resting potentials is due to gradients of ions established by
    „pumps‟ (or ATPases), as well as secondary active transport molecules, that produce
    concentration gradients for various ions – most importantly Na+ and K+. As you will
    see shortly, such concentration differences, combined with the selective permeability of
    the membrane to certain ions (i.e., varying numbers of open ionic channels of different
    types) are the cause of the resting membrane potential. In addition, changes in the
    membrane permeability to various ionic species are the cause of changes in membrane

    By itself, the resting potential of a cell is often of relatively little use. However,
    excitable cells don‟t just have resting potentials. Instead, they use the ionic gradients
    set up by pumps and secondary active transporters to produce changes in membrane
    potential for such purposes as signaling and communication within an individual cell
    and from one cell to its neighbors. Such signals include action potentials and synaptic
    potentials. In addition, changes in membrane potential have become stimuli that can
    themselves alter the behavior of certain ionic channels; these channels (e.g., voltage-
    gated Na+, K+ and Ca2+ channels) can further change membrane potential and initiate
    complex cellular functions, such as long distance signaling via action potentials, the
    initiation of synaptic transmission, the secretion of hormones and the initiation of
    contraction in muscle.

    The clinical importance of such events is obvious. And understanding of what can go
    wrong in such processes first involves understanding their normal operation. This
    includes not only the study of the behavior and function of channels and receptors, but
    also the basic phenomenon that underlie the electrical events of the membrane.

4. In order to understand the origins of the cell membrane resting potential, we will start
   by considering the “concentration cell” presented below (Figure 2.1):
     FIGURE 2.1

             K+                        K+

             Cl-                       Cl-

                i                      e

K+ and Cl- are at higher concentrations in the left hand (side i) compartment than in the
right hand compartment (side e). Osmotic forces are not considered in this example.
However, in each compartment (e or i) the concentration of K+ and Cl- are the same –
thereby preserving electroneutrality. Now consider a situation in which the membrane
separating the two compartments (dashed line) abruptly becomes permeable to K+ but
not to Cl-. What will happen?

a.   K+ will flow down its concentration gradient from the left hand compartment to the
     right hand compartment, i.e., from side i to side e.

b.   However, as K+ moves from the left to the right a net positive charge will build up
     in the right hand compartment because there will be more positive ions (K+) there
     than negative ions (Cl-); remember that Cl- can not cross the membrane. Similarly
     a net negative charge builds up in the left hand compartment due to Cl- ions that are
     „left behind‟ by the flux of K+ across the membrane.

c.   Thus the right hand compartment will build up a positive potential relative to the
     left hand compartment. This potential will oppose further movement of K+.

d.   Net flux of K+ will stop when the electrical potential difference (which would like
     to move K+ from right to left) is sufficient to just balance the concentration
     difference (which wants to move K+ from left to right).

e.   Note that in the final equilibrium situation there will be an excess of positive charge
     on side e, and an excess of negative charge on side i. A “double layer” of ions is
     formed at the membrane, with an excess of positive ions on the side with positive
     potential and an excess of negative ions on the side with negative potential. See
     Figure 2.2. Everywhere else the solutions are electroneutral.
5. Two obvious questions arise: 1) How large will the potential difference be? and 2) How
   much charge (K+) must cross the membrane to establish this potential difference?

    The answer to the first question depends on the concentration difference for K+ on the
    two sides of the membrane. The answer to the second question depends of the
    membrane capacitance (and membrane area). Both of these issues will be considered

6. The Nernst Equation: In the simple case illustrated in Figure 2.1 (where only one ion is
   permeable), the Nernst Equation describes the potential that will develop across the
   membrane on the basis of the ionic concentration difference. The Nernst equation is

                                61mV log [ X ]e
                                  z      [ X ]i
    where z is the valence of the ion (+1 for K+, -1 for Cl-, +2 of Ca2+, etc.). [X]e is the
    concentration of ion X (here K+) on the external side of the membrane (side e) and [X]i
    is the concentration of ion X on the internal side of the membrane (side i). YOU
    SHOULD MEMORIZE THIS EQUATION. Its importance will become clearer as
    these lectures proceed. One thing that the Nernst equation tells you is how far a
    particular ionic species is from electrochemical equilibrium. It is also very useful in
    seeing why changes in the concentrations of various ions have particular effects on the
    membrane potential (e.g., hyperkalemia – which is elevated K+ concentration in the
    extracellular fluid).
        Thus if [K+]i is 100 mM and [K+]e is 10 mM, then in the example illustrated in Figure
        2.1, the membrane potential at equilibrium will be Vm = Vi – Ve = -61 mV. Note that
        if the membrane had been permeable to Cl- rather than K+, then with [Cl-]i = 100 mM
        and [Cl-]e = 10 mM, then the membrane potential would have been +61 mV (the
        difference is due to the valence, z, which is +1 for K+, but –1 for Cl-). However, note
        that in real cell [Cl-]e is usually much greater than [Cl-]i (unlike this example). What is
        the equilibrium potential for Ca2+ if [Ca2+]e = 10 mM and [Ca2+]i = 1 mM? Remember
        that for Ca2+ the valence, z, is +2. Answer: +30.5 mV.

        It is important to recognize that the Nernst equation does not always describe the actual
        membrane potential (as was the case in the simplified situation considered above) – in
        fact, there are few real physiological situations in which this will be the case, because it
        is rarely (if ever) the case that a cell membrane is only permeable to one species of
        ions. Instead, the Nernst equation describes the electrical potential that is needed to
        balance the concentration gradient for a particular ionic species. This is called the
        equilibrium potential, and is denoted by the letter E (e.g., EK or ENa) to distinguish it
        from the actual membrane potential which is denoted by the letter V (Vm).


What is Capacitance?

Capacitance is a constant for any membrane. It is defined mathematically as:

                                           C = Q/V

In laymen’s terms this may be stated as “Capacitance (C) is defined by the amount
of charge (Q, i.e. ions) which line up along the membrane in at a given voltage (V).”
(see Figure 2.2). Recall that in cell, the voltage is set by the concentration
gradients of ions across the membrane (and their permeabilities). If we are
talking about a single ion, the voltage is defined by the Nernst equation, that is, it’s
the amount of electrical energy needed to exactly counteract the tendency of ions
to move down their concentration gradient. Therefore, physiologically speaking,
voltage is the independent variable in the equation above. The charges line up in
reaction to changes in it.
        The capacitance will become more important to you when we talk about
action potential propagation in Lecture 3. We will talk more about the role of
voltage in determining the number of charges which line up along the membrane
there. As far as I am concerned, you only have to know the relationships
defined in the lecture material (e.g. Figures 2.3 and 2.4). This explanation,
like all of the commentaries unless otherwise noted, is strictly FYI.

      Capacitance (i.e. the charges lined up along the membrane opposite one
another) can be thought of as a balance between two forces. The more ions of
opposite charges are attracted to each other across the membrane, the more
charges will line up along the membrane opposite each other. So this force tends
to increase the capacitance. On the other hand, when ions of like charge line up
next to each other on the same side of the membrane, they will repel each other.
So we have the tendency of the ions of opposite charge to form sandwich with the
membrane in between opposed by the tendency of ions of like charge to keep each
other away from the membrane.
       Capacitance increases with membrane area. This is because there is more
“room” for ions of like charge to line up along the membrane. Two ions don’t have
to be as close to one another to be there. As a result, the attraction to the
opposite charges on the other side of the membrane “wins” and you can get more
total charges to line up.
       Capacitance also increases with decreasing effective membrane thickness.
When the ions of opposite charge are brought closer together, the force of
attraction increases. As a result, more ions line up across from one another in a
given membrane area.

                                                                   = Repulsive force
                                               +                   = Attractive force

                +                  +                   +                      +

                -                  -                   -                      -

            -                                                             -
   7. Still considering Figure 2.1, it is reasonable to also wonder just how much K+ – how
      much charge – will have to cross the membrane to establish this potential. The answer
      to this depends on the size of the membrane and on the membrane capacitance. The
      specific capacitance of a typical cell membrane is usually very nearly 1 F per cm2 of
      membrane area (F = microFarad). The total membrane area of cells varies enormously
       from the smallest cells in the body (e.g., red blood cells) to the largest cells such as
       skeletal muscle fibers. As an example, however, we will consider a spherical cell with
       a diameter of 100 m (which could represent a large neuron). If the cell surface is
       smooth (which usually is not the case), then the total surface area would be about
       30,000 m2 (3X10-4 cm2), and the membrane capacity would be about 300 pF (3X10-4
       F, pF = picoFarad = 10-12 Farad). It can be easily be shown (see below, but you are
       not responsible for such a calculation) that for such a cell to establish a membrane
       potential difference of 61 mV it is necessary that about 108 ions must cross the
       membrane. However, if a total ionic concentration of 300 mM is assumed (which is
       approximately normal), there are a total of nearly 1014 ions inside of the cell. Thus, in
       this example, only about 0.0001% of the ions present need to cross the membrane in
       order to establish a 61 mV potential difference.

Commentary: How was the above figure arrived at? Note that you are NOT responsible for
the following calculation; it is only presented here for those of you who might be interested.
Charge, potential and membrane capacitance are related by the following simple equation that
many of you are probably already familiar with:
                                            Q = CV
Where Q = charge, C = membrane capacitance and V = membrane voltage.
For C = 300 pF and V = 61 mV, Q = CV = 1.83x10-11 Coulomb. Noting that the charge on a
single monovalent ion is 1.6x10-19 Coulomb, it follows that 1.83x10-11/1.6x10-19 = 1.14x108
ions must cross the membrane to establish a potential difference of 61 mV.
Cell volume is easily calculated for a spherical cell (100 m diameter) by: Volume of cell =
(4/3)r3 = 5.2x105 m3 = 5.2x10-10 Liter. With an ionic concentration of 300 mM/L (and
remembering that one mole of solute contains 6x1023 molecules) it is easy to calculate that
there will be 0.3x6x1023 = 1.8x1023 ions per liter of solution. Since the cell volume is 5.2x10-10
Liter, the total number of ions is 1.8x1023 x 5.2x10-10 = 9.4x1013 ions.
Thus only somewhat more than 0.0001% (1.14x108/9.4x1013) of the total ions present must
cross the membrane to establish the potential difference in this example. Note that if K+ makes
up half of the intracellular charge – 150 mM – then the amount of K+ that crosses the
membrane is about 0.0002%.
It is also worth recognizing that during a typical nerve or skeletal muscle action potential, it
can be roughly estimated that the number of ions that cross the membrane (Na+ going into and
K+ moving out of the cell) is typically in the range of (very roughly) 20-50 times greater than
the number of ions that need to move to establish the resting potential. In our 100 m diameter
cell, this still amounts to only a relatively small change in concentrations – e.g., intracellular
Na+ concentration might increase by about 0.02 – 0.05% (from about 15 mM to perhaps 15.05
mM). So it is clear that several hundred action potentials could fire without greatly changing
ionic concentrations even without the restorative power the Na+/K+ pump.

       Electroneutrality is only violated in the immediate vicinity of the membrane. Ions that
       cause the transmembrane potential form a „double layer‟ on either side of the
       membrane as illustrated in Figure 2.2, with positive ions on one side of the membrane
       and negative ions clustering on the other side of the membrane. Everywhere else,
    electroneutrality holds (i.e., the number of positive and negative charges in the „bulk
    solution‟ both intracellularly and extracellularly are equal).

    Commentary: As already noted, the number of ions that must cross the cell membrane
    to establish the resting potential is only a very small fraction of the total number of ions
    inside and outside the cell. In fact, diagrams like Figure 2.2 exaggerate the density of
    ions that are in imbalance on either side of the membrane. Considering the same 100
    micron diameter spherical cell that was described above: Note that if K+ ions were
    packed side by side at the surface of the membrane, 1 m2 could accommodate about
    1.1x107 ions. However, in our example, there is at the membrane (on side e) only 3800
    excess K+ ions per m2. So, individual excess K+ ions (if they were spaced in a neat
    square array) would be separated from each other by 0.016 m – quite a distance when
    it is considered that the diameter of a K+ ion is only 0.0003 m.

8. A few more comments about
   membrane        capacitance:       The
   relationship    between     membrane
   potential and charge is simple and it is
   linear. For any given membrane, the                Capacitance=slope
   more charge that crosses the
   membrane, the larger will be the
   resulting potential difference. See
   Figure 2.3.

    FIGURE 2.3 Membrane potential
    versus charge. Which membrane has
    a higher capacitance?       Answer:
    membrane 1. What might be different
    between the two membranes?
    Answer: area, membrane thickness.

                                                                FIGURE 2.3

    Capacitance and Membrane Thickness and Area: Larger cells have more membrane
    area and therefore a higher membrane capacitance. In general, membrane capacitance
    is linearly related to membrane area. A thicker membrane will have less capacitance
    than a thinner membrane. See Figure 2.4. Of course, in real life membrane thickness
    does not vary by any significant amount from one cell to the next. However, myelin
    effectively increases membrane thickness and thereby reduces membrane capacitance.
    Myelin is a wrapping of layer after layer of cell membranes (oligodendrocites or
    Schwann cells) that surrounds many nerve cell axons. The ultimate affect is simply like
    having a very thick membrane, i.e., capacitance is reduced and therefore less charge is
    needed to change the membrane potential (see my next lecture for further details).
                                 TAKE HOME MESSAGE
                                        up to this point

1. Know the basic rules that govern the movement of ions in solution and more importantly
   across membranes.
2. Know the definition of transmembrane potential, i.e., Vm = Vi – Ve.
3. Understand the concept of electroneutrality (i.e., an equal number of positive and
   negative charges in the bulk solutions on either side of the membrane). Recognize that
   electroneutrality is violated in the immediate vicinity of the membrane whenever a
   membrane potential is present, but that the overall violation is a very small fraction of the
   total charge on either side of the membrane.
4. Understand the forces that bring an ion into equilibrium across a selectively permeable
   membrane (i.e., a membrane that is only permeable to some of the ionic species that are
   present). Recognize that electrochemical equilibrium is reached when the chemical
   forces and electrical forces are in balance.
5. Memorize the Nernst equation, and don‟t forget the importance of the valence, z, in this
   equation (both in determining the sign and magnitude of the equilibrium potential).
6. Understand the difference between equilibrium potentials (denoted by the letter E) and
   the membrane potential (denoted by V). Equilibrium potentials are specific to each ionic
   species and are a measure of the electrochemical energy stored by concentration
   difference across the membrane for that ion. The membrane potential is the actual
   potential difference that exists across the cell membrane at any point in time, and is rarely
   the same as the equilibrium potential for any ion (exceptions to this are cases in which the
   membrane is permeable to an ion but is not actively transported; in many cells at rest this
   is true of chloride).
7. Understand the basic concept of membrane capacitance, which is simply a measure of
   how much charge is needed to create a given membrane potential. Understand the effects
   of membrane area and membrane thickness on capacitance (i.e., the larger the area, the
   larger the capacitance and the greater the thickness – or „effective‟ thickness of the
   membrane – the lower the capacitance).
8. Understand the relationship between cell, or cellular compartments, size and the
   concentration changes that occur to produce the resting potential or an action potential.
   In particular, understand that as cell (or cellular compartment) size decreases, the
   magnitude of concentration changes (or imbalances in charge) increases. The most
   extreme example of this that we will consider is the transverse tubular system of skeletal
   muscle, which – due to its very small volume and large surface area – has relatively large
   fractions of ions cross the membrane to produce the resting potential and can experience
   relatively large changes in ionic concentrations as the result of a single action potential.
   This has important clinical consequences in certain diseases such as myotonias.
(Recall that you are not responsile for Clinical Correlations unless the material
relates to block cases or cases presented in workshop.)
9. The example of the „concentration cell‟ presented above assumed that the membrane
   separating the two compartments was completely impermeable to all solutes except K+.
   It is very important to recognize that at rest many cell membranes are primarily
   permeable to K+, but that none are exclusively permeable to this ionic species. Thus, by
   itself, the Nernst equation is usually not able to accurately predict the resting potential
   of a cell, much less the potential changes that will occur when the permeabilities of the
   membrane to certain ionic species change. For this, other, more complicated equations
   are required. Two equations are in common usage to give approximations to what can
   be expected when the membrane is permeable to multiple ionic species. These are the
   Goldman-Hodgkin-Katz (GHK) equation (also called the constant field equation) and
   the chord conductance equation. These are briefly presented here for several reasons.
   In the first place, they may help you to understand what happens to membrane potential
   when more than one ionic species can cross the membrane (but words can also provide
   you with this understanding). In the second place, one or both of these equations
   sometimes shows up in the board exams (and they are both usually covered in board
   review books); so some familiarity with them (and especially the messages that they
   convey) is certainly worthwhile. However, I will never ask you to use either equation
   for a computation on an exam question in this course. What you are responsible for is
   the underlying meaning of these equations. Thus some practice with the equations may
   be useful.

    The Goldman-Hodgkin-Katz (GHK) equation (also called the “Constant Field
    Equation”) is as follows:

                       [K  ]e PK  [Na  ]e PNa  [Cl  ]i PCl
          Vm  61mV log 
                       [K ]i PK  [Na  ]i PNa  [Cl  ]e PCl
    Here it has been assumed that the membrane is permeable to only K+, Na+ and Cl- (with
    permeabilities defined by P; more complex versions of this equation are possible that
    also take into account other ionic permeabilites). This equation is one way of
    understanding the competing forces that determine the resting potential when (as is
    essentially always the case) more than one type of ion is permeable. It is also useful in
    understanding changes in membrane potential that occur during synaptic activity and
    action potentials. It is an approximation, but it is a reasonable and useful
    approximation. You are not going to be held responsible for making computations
    using the GHK equation. However, you should have a qualitative and semi-
    quantitative understanding of what it means. This equation indicates that the most
    permeable ions will have the greatest effect on the transmembrane potential. For
    example note that if the permeabilities to Na+ and Cl- are zero, then the GHK equation
    reduces to the Nernst equation for K+. Similarly, if the permeability of the membrane
    to K+ is much greater than that for Na+ and Cl-, then the equation indicates that the
    membrane potential will be very near the equilibrium potential for K+.
Another equation that describes the membrane potential when several ionic species are
permeable is called the “Chord Conductance Equation”. To understand this
alternative equation we must first introduce the concept of membrane conductance.
The conductance of the membrane to any given ionic species is simply a measure of
how easily this ion can cross the membrane. Thus it is essentially analogous to the
membrane permeability for this ionic species (there are differences, but you don‟t have
to worry about them in this course). More open channels means more conductance;
similarly, open channels that allow more ions to cross per second at any particular
potential also mean a higher membrane conductance. The membrane conductance to a
particular ionic species, X, is denoted by gX (e.g., gK, gNa). The current, iX, through the
membrane for an ionic species, X, is then defined by:

                                   iX = gX(Vm – EX)
where Vm is the membrane potential and EX is the equilibrium potential (as defined by
the Nernst equation) for the particular ionic species. Note that by convention, positive
charges leaving a cell (or negative charges entering the cell) are designated “positive”
current. Vm – EX is referred to as the driving force, (also called the electrochemical
potential difference, or EPD) since it is this potential difference that drives the
particular ionic species across the membrane through open ionic channels. Note that
for any particular conductance, the membrane current – i.e, the net number of ions
crossing the membrane per second – will be zero if Vm = EX, and will linearly increase
as Vm moves away from EX.

When the membrane is simultaneously permeable to more than one ionic species, a
steady state situation (i.e., a situation in which the membrane potential is constant –
such as the resting potential) can only occur if the sum of all of the ionic currents is
zero (i.e., there is no net ionic current). If this was not the case – i.e., if there was a net
ionic current – then the ionic current would necessarily be charging the membrane
capacitance and therefore changing the membrane potential. For example, if the
membrane is permeable to Na+ and K+, then it is necessary at rest that iNa + iK = 0. So,
gNa(Vm-ENa) + gK(Vm-EK) = 0. And solving for Vm this means that:

                Vm = {gK/(gNa+gK)}EK + {gNa/(gNa+gK)}ENa
This is the chord conductance equation when the membrane is only permeable to K+
and Na+. Similar expression can be derived for any number of ionic species, so that a
generalized equation is possible. For ions X, Y, Z, … and with gT defined as
gX+gY+gZ…, then Vm is given by:

                 Vm = (gX/gT)EX + (gY/gT)EY + (gZ/gT)EZ + …
This equation has the same „take home message‟ as the GHK equation, namely that the
ionic species with the largest conductance has the greatest effect on the membrane
potential. In a situation where 2 ionic species have essentially comparable
conductances, the membrane potential will essentially „split the difference‟ between the
equilibrium potentials of these ions.
At rest (i.e., when the cell membrane is at its resting potential), most cells have a
relatively high permeability (conductance) to K+ and a much lower permeability to Na+.
Many cells (e.g., skeletal muscle cells) also have a relatively high resting permeability
to Cl-. Note that I have used the term “relatively” because for most excitable cells the
total membrane permeability to all ions at rest is much less than the total membrane
permeability during all or a part of the action potential (see lectures 3 and 4).

In any case, since the membrane at rest is more permeable to K+ than to Na+, the resting
membrane potential is much closer to EK than to ENa. Typical intracelullar and
extracellular concentrations of some important ions are listed in Table 2.1.
Approximate equilibrium potentials (on the basis of the Nernst equation) are also listed
in this table, as are the relative resting permeabilities in a typical skeletal muscle fiber.


Ionic          Typical              Typical               Typical              Typical
Species        Intracelluar         Plasma                Eqiulibrium          PX/PNa*1
               Concentration        Concentration         Potential

 Na+           10-15 mM             135-145 mM            +60 mV                  1
 K+            140 mM               4 mM                  -95 mV                 50
 Cl-           4-15 mM              98-106 mM             -50 to –85 mV          50
 Ca2+          100 nM *2            2.0 –2.6 mM           +130 mV              < 0.1

Notes:       The proportional values presented here are approximately correct for skeletal
          muscle fibers at rest. Many other ratios exist for other tisues.
             The value of 100 nM (100x10-9 Molar = 0.1 M) is approimately appropriate
          for a „resting‟ cell; higher concentrations of intracellular Ca2+ occur in many
          situations such as when muscles are contracting.

It is important to recognize that the concentration differences listed in Table 2.1 for
different ionic species usually result from primary and secondary active transport
processes (e.g., the Na+/K+ pump, the Ca2+ pump, the Na+/Ca2+ exchanger). These
transport molecules directly or indirectly use metabolic energy to establish and
maintain the ionic concentration differences described.

It is also important to realize that cells contain a high concentration of impermeable
anions (mostly proteins and amino acids) that essentially never cross the cell
membrane under normal circumstances. These are generally denoted by A -, with [A-]i
 100 mM being typical The interstitial fluid contains only a very small concentration
of such solutes (proteins); plasma contains a larger concentration than the interstitial
fluid (i.e., plasma proteins), but this is still much less than the concentration that occurs
inside of cells.
    Chloride (Cl-) deserves some extra attention. In many (probably most) cells, chloride is
    not actively transported (exceptions certainly exist, such as the Na-K-2Cl cotransporter
    that is important to the function of many epithelia, as well as some other cells, and
    transports Cl- against its electrochemical gradient). However, for now, let us consider a
    situation in which Cl- is NOT actively transported; in many cells this is the case, and in
    many others it is a good approximation. But also consider that the membrane is
    permeable to Cl-; further consider that the cell is at rest (no action potentials or synaptic
    potentials). In this situation, what will be the equilibrium potential for chloride? The
    answer is simply that the chloride equilibrium potential (ECl) will be equal to the
    membrane potential (Vm), i.e., ECl = Vm. In this case the concentration gradient for Cl-
    is NOT helping to establish the membrane resting potential. Instead, the resting
    potential is determining the concentration gradient for Cl-. The resting potential itself is
    determined by the permeability of the cell membrane to ions that are moved by active
    transport (e.g., Na+ and K+). Chloride is simply assuming a transmembrane
    concentration difference that is appropriate for this transmembrane potential.

    However, this does NOT mean that the membrane‟s resting permeability to chloride (or
    changes in this permeability) can not be important in non-steady state situations. This
    is because it normally takes a long time for ionic concentrations to significantly change
    across the cell membrane. As a specific example, consider a nerve cell with a resting
    potential of –70 mV and an external concentration of Cl- of 100mM; based on the
    resting potential and the external chloride concentration it is easy to calculate that the
    internal Cl- concentration must be about 7.1 mM. (Aside: this is the internal
    concentration of Cl- that is required for ECl to be –70 mV based on the Nernst equation).
    If the membrane potential of this cell was abruptly to change to –50 mV, then it is also
    easy to calculate that in the steady state [Cl-]i will eventually increase to about 15.1
    mM. But for this change to become complete could require many seconds or even a
    few minutes (the time depends on the membrane‟s permeability to chloride and on the
    size of the cell). So if the membrane potential transiently changes, then the
    membrane‟s permeability to Cl- will try to hold the potential at its resting value.
    Similarly, transient increases in the membrane permeability to Cl- will also help to hold
    the membrane near its resting potential (for example, thereby opposing excitatory
    postsynapitc inputs).

10. It is also important to realize that (like several other active transport molecules) the
    Na+/K+ pump is electrogenic in nature. This results from the fact that the Na+/K+ pump
    moves 3 Na+ ions out of the cell and 2 K+ ions into the cell for every molecule of ATP
    that is split. Thus there is a net movement of charge during every cycle (one positive ion
    out of the cell). This produces a potential difference that contributes to the cell
    membrane resting potential. This contribution has been ignored in the GHK and chord
    conductance equations presented above. This is generally reasonable, since the Na+/K+
    pump typically only contributes 10 mV or less to the resting membrane potential.
    Nevertheless, its contribution should not be forgotten, and can be greater in some cells.
                              TAKE HOME MESSAGE
                               For the rest of Lecture 2

1. Remember that you are not going to ever to make computations using the GHK or
   the full form of the chord conductance equation on exams (but making a few
   computations with these equations couldn‟t hurt, since they are included in most
   board review books). However, some aspects of these equations may show up in
   simple fashion on the exam

2. However, you are responsible for the messages that these equations describe. These
   messages are fairly simple:
      a. If one ionic species is much more permeable than all others, then the
         membrane potential will closely approach the equilibrium potential for this
      b. If the membrane is roughly equal in permeability (conductance) to two ions
         (as is frequently the case for some ionic channels and receptor/channels) then
         the membrane potential will roughly split the difference between the
         equilibrium potential for these ions.
      c. When the membrane is permeable to several different ions (e.g., via different
         ionic channels – as is often the case in cardiac muscle), then only the actual
         equations can give a good approximation of the membrane potential. But
         reasonableness still applies: the most permeable ions have the greatest effect,
         but all permeable ions must be considered to arrive at accurate predictions.

3. Chloride is an ion that is (for the resting membrane) often in equilibrium across the
   cell membrane. This is because the membrane is permeable to Cl-, but this ion is not
   actively transported in many (or most) cells.

4. Nevertheless, chloride permeability (and changes in the membrane permeability to
   Cl-) can have important affects of cellular function and cell membrane potential.
   This is because changes in membrane potential (e.g., action potentials or synaptic
   potentials) are usually of relatively short duration, while the time required to change
   the transmembrane chloride concentration is generally much longer. So if a cell
   moves away from its resting potential, then chloride channels (with E Cl roughly
   equaling the resting membrane potential) will „try‟ to hold the membrane potential
   near its resting level. This is of great importance in understanding many inhibitory
   synapses in the CNS. It is also important in many other types of tissues (e.g., the

5. Remember that the Na+/K+ pump is electrogenic, because it moves more Na+ ions
   out of the cell than K+ ions into the cell (by a ratio of 3 to 2). Because of this, this
   pump makes a relatively small contribution to the resting potential in most cells. It
   is also important to cell volume regulation.

Note that this and other Membrane Physiology quizzes are simply provided to you to help you to
test your understanding of the content of each lecture. They are not substitutes for studying the
lecture notes (including Objectives and Take Home Messages), Practice Exam Questions or
other material intended to guide your learning. These questions are not intended to reflect the
style of level of difficulty of questions on the Midterm.


1.     A membrane separates two compartments each of which contains a KCl solution as
       shown in the following figure. Side i contains a higher concentration of both K+ and Cl-
       than side e. The volume of the compartments is far larger than that of any cell in the
       human body. If the membrane abruptly becomes permeable to K+ but remains
       impermeable to Cl-, then once a new steady state is reached:

       a.     side i will have a negative potential relative to side e
       b.     side i will have more Cl- ions than K+ ions
       c.     side e will have more K+ ions than Cl- ions
       d.     the difference in concentration of K+ and Cl- on each side is likely to be
              immeasurable by chemical techniques.
       e.     All of the above are true

2.     If the extracellular concentration of Na+ is 15 mM and the intracellular concentration of
       Na+ is 150 mM, then the equilibrium potential for Na+ is:

       a.     –61 mV
       b.     –30.5 mV
       c.     0     mV
       d.     +61 mV
       e.     +122 mV

3.     If the extracellular concentration of Ca2+ is 2 mM (2x10-3 M) and the intracellular
       concentration of Ca2+ is 200 nM (2x10-7 M), the equilibrium potential for Ca2+ is:

       a.     –122 mV
       b.     –61 mV
       c.     0    mV
       d.     +122 mV
       e.     +244 mV
4.   In many cells, the resting membrane has a relatively high permeability to chloride, but
     chloride is passively distributed across the membrane (i.e., it is not actively transported).
     In this situation when the cell membrane potential (Vm) has been at its resting level (here
     say this is –70 mV) for a long period of time:

     a.     ECl is equal to Vm
     b.     ECl must be less negative than Vm
     c.     ECl must be more negative than Vm
     d.     the extracellular concentration of Cl- must be less than its intracellular
     e.     both a and d are correct

5.   An important message from the GHK or the constant field equation is:

     a.     All permeable ionic species contribute about equally to the transmembrane
            potential at all times
     b.     Impermeable ions contribute as much to the resting membrane potential as do
            permeable ions
     c.     The cell membrane potential will be closer to the equilibrium potential of a highly
            permeable ionic species than to that of an ionic species with low membrane
     d.     When the cell membrane is at its resting potential, there will be a continuous
            movement of net negative charge into the cell
     e.     None of the above are correct

6.   The equilibrium potential of a particular ionic species is:

     a.     always equal to the membrane potential
     b.     directly dependent on the membrane permeability for that ionic species
     c.     the membrane potential at which there will be no net flux of the ion, regardless of
            its membrane permeability
     d.     a measure of the concentration difference of the ion across the membrane
     e.     both c and d are correct

7.   A membrane is permeable to both Na+ and K+ (and impermeable to other ions). ENa =
     +60 mV and EK = –100 mV. The relative conductance of the membrane to K+ is 15 times
     larger than the membrane conductance to Na+. On the basis of this information, which
     of the following is/are correct?

     a.     the membrane potential is closer to EK than to ENa
     b.     the K+ current is equal to the Na+ current, but of opposite polarity
     c.     the K+ current is larger in magnitude than the Na+ current
     d.     both a and b are correct
     e.     both a and c are correct

8.     If there is an equal concentration difference across the cell membrane for both K+ and Cl-
       (both higher on one side than the other) then these ions will have the same equilibrium

9.     A thicker membrane will have a higher capacitance per unit area than a thinner

10.    In order to establish a resting potential of about –70 mV in a typical size cell, it is
       necessary that about 1% of the ions present within the cell will have to cross the

11.    As the result of an action potential, concentration changes in the lumen of the transverse
       tubular system of skeletal muscle are usually much larger than those across the surface
       membrane of these cells.

12.    The equilibrium potential for K+ is normally identical to the transmembrane resting

13.    Under normal conditions the cell membrane potential will never be more positive than
       the Na+ equilibrium potential.

14.    Under normal conditions the cytoplasmic concentration of Ca2+ is higher than its
       extracellular concentration.

15.    Increased extracellular K+ concentration (hyperkalemia) will cause most cells to have a
       less negative resting potential.

Answers: 1. e, 2. a, 3. d, 4. a, 5. c, 6. e, 7. d
        8. F, 9. F, 10. F, 11. T, 12. F, 13. T, 14. F, 15. T

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