What causes
this voltage?
Penetrate
A microelectrode is a specialized electrode for recording voltages from the inside of cells. It typically consists of a glass tube drawn out to a very fine point and filled with an electrolyte solution such as potassium chloride. The sharp tip of the electrode penetrates through the cell membrane. The blunt end of the electrode is connected by a wire to a voltage measuring instrument.
The Membrane Potential
Instruments for recording voltages from cells have to have a very high input impedance. This means that they can measure the voltage without drawing much current from the cell, and thus without changing the very voltage that they are trying to measure.
-70
All living cells maintain a potential difference across their cell membranes, with the inside usually negative relative to the outside. In nerve cells the value of the resting potential varies between about -40 and -90 mV.
The membrane potential can be measured by penetrating the cell with a microelectrodeA microelectrode is a specialized electrode for recording voltages from the inside of cells. It typically consists of a glass tube drawn out to a very fine point and filled with an electrolyte solution such as potassium chloride. The sharp tip of the electrode penetrates through the cell membrane. The blunt end of the electrode is connected by a wire to a voltage measuring instrument.. This is connected to a specialized voltmeterInstruments for recording voltages from cells have to have a very high input impedance. This means that they can measure the voltage without drawing much current from the cell, and thus without changing the very voltage that they are trying to measure. which measures the potential difference between the inside of the cell and the surrounding environment. This voltage is the membrane potential.
This tutorial describes the cellular mechanisms that generate the resting membrane potential.
Withdraw
Early in the 20th century, Bernstein proposed that the resting membrane potential was due to 3 factors:
the cell membrane is selectively permeable to potassium ions,the intracellular potassium concentration is high,the extracellular potassium concentration is low.This is known as the potassium electrode hypothesis.
In the next section we will consider how these factors would cause a membrane potential with the inside negative.
This will lead to an understanding of the Nernst equation and the ionic equilibrium (reversal) potentials.
Sneak preview - Bernstein was almost, but not completely, right.
Bernstein’s Potassium Hypothesis
previous
contents
next
1. The Membrane Potential
Simple Goldman equation
6. Na-K Exchange Pump
Key points
2. Bernstein’s Potassium Hypothesis
Action potential
7. Unusual Mechanisms
4. The Goldman Equation
Explaining Bernstein
Chloride ions
Steady state
Key Points
Concentration changes
3. The Nernst Equation
Gradient run-down
Contents
Deriving the Nernst equation
Electrogenic pumps
3. Donnan Equilibrium
Using the Nernst equation
Donnan equation
5. Leakage channels
Expaining Donnan
Reversal potential
8. Summary
Frog and squid
+100
mV
-100
0
1. Concentration measurements show cells do indeed have a K⁺ gradient across the membrane in the appropriate direction (high inside, low outside).
2. Flux measurements using radioactive potassium show that the membrane is permeable to K⁺.
3. These facts are consistent with Bernstein’s hypothesis for why there is a membrane potential with the inside negative.
4. Note that it is the side with the high concentration of positive ions that becomes the negative side of the membrane (because the positive ions flow away from the high concentration side). This often causes confusion.
5. Not many ions have to flow to set up the potential, so the concentration on the two sides is virtually unchanged. This will be explained in more detail later; for now just trust me …
6. The potential is set up almost instantly, although the cartoon illustration shows it taking quite some time.
Now let’s quantify this ...
Potassium electrode hypothesis: Key points
K+
+
electrical
gradient
4. When the electrical gradient exactly balances the chemical gradient, equilibrium is achieved. There is no further net movement of ions.
Show
K can flow!
3. The electrical gradient opposes the movement of positively-charged potassium ions.
-
Step
K+
gradient
1. Potassium ions diffuse down their concentration gradient. They carry positive charge with them (K⁺).
Explaining Bernstein 4
HOWEVER, as far as K+ ions are concerned, the two compartments are now connected through pores in the membrane (shown as just a single gap in the diagram).
What happens now?
Reset
2. Positive charge accumulates in the outside compartment, building up an electrical gradient (voltage) across the membrane.
Ca
I suggest you look at the pre-built examples first:
K
[Xout]
VmV =
K⁺
Ca⁺⁺
Cl⁻
Examples
inside
Using the Nernst Equation
Then try it out yourself:
mM
outside
Cl
log10
-58
Note: if the membrane is only permeable to one ion, then the membrane potential will be the same as the Nernst potential. This is approximately true for glial cells, which are almost exclusively permeable to potassium. However, most cell membranes are permeable to more than one ion type, which makes things more complicated. This is explored further in the Donnan and Goldman sections of the tutorial.
Select the ion to which the membrane is permeable, and the intracellular and extracellular concentrations.
z
[Xin]
When the driving force is zero, potassium ions are in equilibrium across the membrane and just move randomly back and forth.
As the electrical gradient builds up, it counteracts the chemical gradient, and the driving force decreases.
The difference between the chemical gradient and the electrical gradient is called the driving force.
In our scenario there is initially a large driving force (shown as an arrow through the pore). This is because the chemical gradient is not balanced by any electrical gradient.
Driving force
Explaining Bernstein 5
The Experimental Situation
Explaining Bernstein 1
We start off with a simple container divided into two compartments.The left-hand compartment represents the inside of a neuron, while the right-hand compartment represents the outside, i.e. extracellular space. The barrier represents the cell membrane.
Assume that both compartments contain only water, and that the barrier is impermeable to allIf the barrier were truly completely impermeable to everything (i.e. a perfect insulator) then the voltmeter inputs would be “floating” and it would probably show random values due to picking up weak electric fields in the environment.
We can avoid this by assuming that the barrier is very slightly permeable to water. ions.
A voltmeter measures the potential difference between them (in-out). Since the two compartments have identical (empty) contents, this is zero (0 mV).
If the barrier were truly completely impermeable to everything (i.e. a perfect insulator) then the voltmeter inputs would be “floating” and it would probably show random values due to picking up weak electric fields in the environment.
We can avoid this by assuming that the barrier is very slightly permeable to water.
Add Some Ions
We now add some KCl to both sides. We put a relatively high concentration (100 mM) into the inside, and a low concentration (10 mM) into the outside.
The KCl dissociates into K⁺ and Cl⁻.
There is now a chemical concentration gradient between the two compartments, but since the membrane is impermeable to K⁺ and Cl⁻, this has no effect on the membrane potential, which remains at 0.
K+ Cl-
gradient
Explaining Bernstein 2
Cl-
Explaining Bernstein 3
Change the Barrier
We now replace the impermeable barrier with a semi-permeable membrane, which is permeable to potassium (K⁺) ions only.
So far as the chloride (Cl⁻) ions are concerned, the two compartments are totally separated by the barrier, and so the concentration gradient in Cl⁻ ions has no effect.
The Nernst equation is fundamental for understanding not only the mechanism generating the resting potential, but also that of action potentials (spikes) and synaptic potentials.
We have seen empirically that equilibrium will occur if a gradient in chemical concentration, that would normally cause an ion to move from the high concentration to the low concentration, is balanced by an electrical gradient that opposes the movement of charge.
Nernst equation
The Nernst equation is a formula that relates the numerical values of the concentration to the electrical gradient that balances it.
electrical gradient
The Nernst Equation
chemical gradient
Electrical energy
Deriving the Nernst equation 1
V
RT
Equilibrium is achieved when the energy released by a particle sliding down a concentration gradient is equal to the energy required to push a charged particle up an electrical gradient.
The equations describing these energies are derived from thermodynamic principles (which we won’t go into …)
Rearranging the equations gives us the Nernst equation.
=
[X2]
δWe
where δWe is the energy change associated with moving n moles of a charged particle with valency z in an electric field of strength V (volts), and F is Faraday's number (C mol⁻¹).
ln
where δWc is the energy change associated with moving n moles from concentration X1 to concentration X2 (mol l⁻¹), R is the gas constant (8.31 J K⁻¹ mol⁻¹), T is the absolute temperature (°K), ln is the natural logarithm (base e).
Chemical energy
z F
n
The Conditions for Equilibrium
δWc
[X1]
(Note it is minus 58)
Simplify
Deriving the Nernst equation 3
However, R and F are constants, and if we assume that T is room temperature (21℃), convert from natural logarithm to log base 10, state V in millivolts rather than volts, make compartment 1 the inside of a neuron and compartment 2 the outside, the equation simplifies to:
V =
The “official” Nernst equation is:
The Nernst equation defines the equilibrium potential for an ion - i.e. the electrical potential that exactly balances a concentration gradient for that ion. This is also called the reversal potential, for reasons that will become clear later ...
Each ion will have its own individual equilibrium potential, depending on its concentration gradient across the membrane.
Using the Nernst Equation
The potassium concentration ratio is 10:1.
The log of 10 is 1, potassium is monovalent and carries a single +ve charge, so the solution to the Nernst equation is simply -58 mV.
10
For a monovalent ion a 10-to-1 inside-to-outside concentration ratio is exactly balanced by a -58 mV membrane potential.
K specific
Example 1 of 5
VmV = -58 log(10) = -58 x 1 = -58
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100
Na
Na specific
46
If we reverse the direction of the concentration gradient, so that the ratio is now 1:10 (i.e. 0.1), then the absolute value of the membrane potential stays the same, but its polarity reverses (the log of 0.1 is -1).
Note that the ion has changed to sodium but the Nernst equation doesn’t care - it is still a monovalent ion. (We assume the membrane is now permeable to sodium.)
460
Example 3 of 5
VmV = -58 log(0.1) = -58 x -1 = 58
The polarity of the Nernst equation depends on the direction of the concentration gradient.
58
Example 2 of 5
VmV = -58 log(460/46) = -58log(10)
= -58 x 1 = -58
The ratio of 460:46 is still 10:1, and the Nernst potential is still -58 mV.
It doesn’t matter what the absolute values of the concentrations are, it is the ratio that counts.
Ca
This is a bit counter-intuitive, but it takes more energy to move two charges in an electric field than just one.
This means that a half-strength field will balance a given concentration gradient involving a divalent ion compared to the field strength required to balance the same concentration gradient involving a monovalent ion.
Example 4 of 5
log(0.1) = -29 x -1 = 29
For a divalent ion (z = 2 in the Nernst equation) the membrane potential is halfThis is a bit counter-intuitive, but it takes more energy to move two charges in an electric field than just one.
This means that a half-strength field will balance a given concentration gradient involving a divalent ion compared to the field strength required to balance the same concentration gradient involving a monovalent ion. what it is for a monovalent ion with the same concentration ratio.
The ratio is still 1:10, but for Ca²⁺ the Nernst potential is 29 mV.
Ca specific
29
2
Example 5 of 5
log(0.1) = 58 x -1 = -58
We are back to a monovalent ion, but the chloride ion carries a negative charge (z = -1), so the membrane potential is reversed compared to an ion with the same concentration gradient but a positive charge.
Cl
The ration is still 1:10, but for Cl⁻ the Nernst potential is -58 mV.
Cl specific
-1
The ions flow in the opposite direction if they are negatively charged, like chloride.
-75
Positive
If the membrane potential is at the equilibrium potential of an ion (in this case -75 mV), the ions move randomly back and forth across the membrane through any open channels selective for the ion.
If the membrane potential is below the equilibrium potential, then positivelyThe ions flow in the opposite direction if they are negatively charged, like chloride. charged ions like potassiumn will flow into the cell, but if the membrane potential is above the equilibrium potential, the ions will flow out of the cell.
Thus the direction of current carried by an ion reverses when the membrane potential is on either side of its equilibrium potential, which is why this is also called the reversal potential.
Negative
Equilibrium = Reversal Potential
Reversal Potential
-50
A battery is inserted to make the inside of the cell more positive, taking the membrane potential above the potassium equilibrium potential.
The electrical gradient is now weaker than the chemical gradient, there is an outward-directed driving force, and positively-charged potassium ions flow down the concentration gradient out of the cell.
This flux will continue unless and until the concentration gradient decreases to achieve a new Nernst equilibrium potential.
Above equilibrium potential
Reversal Potential
A battery is inserted to make the inside of the cell more negative, taking the membrane potential below the potassium equilibrium potential.
The electrical gradient is now stronger than the chemical gradient, there is an inward-directed driving force, and positively-charged potassium ions flow up their concentration gradient into of the cell.
This flux will continue unless and until the concentration gradient increases to achieve a new Nernst equilibrium potential.
?
If the membrane potential of a cell is caused by potassium ions moving down their concentration gradient, does this alter the internal concentration of potassium?
In other words, how many ions have to leave the cell in order to set up the equilibrium potential?
How Many Ions Move?
Concentration Changes 1
K⁺
leaves
The Membrane is a Capacitor
Concentration Changes 2
This question can be approached by treating the cell membrane as a capacitor, in which the intracellular cytoplasm and extracellular fluid are the conducting electrolytes, and the phospholipid component of the cell membrane is the insulating dialectric.
The typical value of capacitance for a biological membrane is 1 µF cm⁻². For a spherical cell with a diameter of 50 µm the surface area (4πr2) is 7.8 x 10-5 cm², so the capacitance is 78 pF.
If the membrane potential is -75 mV, the charge on the capacitor is:
Q = VC = 75 x 10-3 × 7.8 x 10-11
Thus Q = 5.9 x 10-12 coulombs.
One K⁺ ion has a charge of 1.6 x 10-19 coulombs, so the number of ions that move down the concentration gradient is about
37,680,000
i.e. about 38 million.
Concentration Changes 3
38 million K⁺ions leave
About 38 million Ions Leave
Concentration Changes 4
0.006% of K⁺ions leave
Does this Alter the Concentration Gradient?
A cell with diameter 50 µm has a volume (4πr³/3) of 2.6 x 10-12 litres and its intracellular K⁺ concentration is about 400 mM, so it contains 1.04 x 10-12 moles of K⁺ ions. One mole consists of 6.02 x 1023 ions (Avogadro's number), so the cell contains about 6.2 x 1011 ions. The loss of 38 x106 ions represents a concentration change of about
0.006%
So the answer is HARDLY AT ALL.
Note: the external volume is relatively huge, so the change in external concentration is infinitesimal.
Nernst equation: Key points
1. The Nernst equation defines the electrical potential across a membrane that will balance a particular chemical concentration gradient of an ion. So if the membrane potential is at the Nernst value, there will be no net movement of that ion across the membrane, even if the membrane is permeable to the ion.
2. This potential is called the equilibrium or reversal potential for that ion. It depends on the concentration gradient (ratio) and the valency of the ion (and the temperature), but does not depend on the degree of permeability of the membrane to the ion, nor on the absolute concentrations.
3. Each type of ion will have its own equilibrium potential, and this is likely to be different to that of other ion types.
4. If a membrane is permeable to only ONE type of ion, then the membrane potential will automatically move to the equilibrium potential for that ion.
5. Relatively few ions have to move to set up the potential, so the concentration gradient is not significantly perturbed in achieving equilibrium.
However, what if the cell membrane is permeable to more than one type of ion …?
The intracellular composition is thus:
K⁺Cl⁻ + K⁺A⁻
This can result in a Donnan equilibrium, which both maintains opposite K⁺ and Cl⁻ gradients across the membrane, and sets up a resting potential.
The Donnan equilibrium applies to ions whose distribution is passive and unregulated, i.e. there are no metabolic pumps maintaining the intracellular concentrations at fixed values.
Donnan Equilibrium
Many cells have membranes with quite a high permeability to both potassium and chloride ions. In the absence of anything else, this would prevent the establishment of the membrane potential, because Cl⁻ would simply move down the concentration gradient along with the K⁺, and counteract the positive charge. The concentrations would just equilibrate across the membrane.
However, most cells also have a relatively high intracellular concentration of large molecular weight anions A⁻ (e.g. negatively charged proteins).
There is no guarantee that equilibrium can be reached. In many cells metabolic pumps maintain intracellular ion concentrations within fairly narrow ranges. In these cases the resting potential is a steady state situation, rather than a chemical equilibrium, and the Donnan rule does not apply. This is explored further in the Goldman section of the tutorial.
If the membrane is permeable to both K⁺ and Cl⁻ , both will try to move across the membrane until their concentration gradients are balanced by the membrane potential as given by the Nernst equation.
There can be only one membrane potential across a patch of membrane at any one time, so ifThere is no guarantee that equilibrium can be reached. In many cells metabolic pumps maintain intracellular ion concentrations within fairly narrow ranges. In these cases the resting potential is a steady state situation, rather than a chemical equilibrium, and the Donnan rule does not apply. This is explored further in the Goldman section of the tutorial. equilibrium is reached:
[K⁺in]
This is the Donnan rule of equilibrium.
[Cl⁻out]
[K⁺out]
[Cl⁻in]
(Note the K-Cl gradient inversion due to the opposite charge of K⁺ and Cl⁻.)
1 F
Thus at equilibrium the product of the concentrations of diffusible ions on one side of the membrane equals the product of the concentrations of the diffusible ions on the other:
Losing the common factors gives:
The Donnan Equation
= V =
[K⁺in] × [Cl⁻in] = [K⁺out] × [Cl⁻out]
in
If the membrane is permeable to both K⁺ and Cl⁻, and no impermeant molecules are involved, then equal numbers of K⁺ and Cl⁻ ions cross the membrane, until the concentrations are equal on either side.
Because exactly equal numbers of positive and negative charges move across the membrane, no membrane potential is set up.
The final state obeys the Donnan rule, but it is not very interesting because there is no concentration gradient and no membrane potential.
out
Explaining Donnan 1
Run-Down!
Now consider the situation where some intracellular Cl⁻ ions are replaced by the impermeant anion A⁻. The K⁺ and Cl⁻ ions can still flow down their concentration gradients and leave the cell, but A⁻ cannot pass through the membrane.
In the diagram each symbol represents 10 mM of substance - e.g. there are 12 K⁺s giving a total of 120 mM.
We start off with everything inside the cell, nothing outside.
What happens now?
K+ gradient
A⁻
Explaining Donnan 2
Cl- gradient
Impermeant Intracellular Anions
Details
So more K⁺ leaves the cell. Its positive charge attracts Cl⁻ ions, and so more Cl⁻ moves out of the cell in order to maintain space-charge neutrality (it has to be Cl⁻ because A⁻ cannot cross the membrane). This means that the Cl⁻ gradient now starts to reverse from its original condition.
K⁺ is still "winning" the osmotic battle at this stage, because its gradient (85:35) is greater than the reverse Cl⁻ gradient (25:35). So K⁺ will continue to leave the cell, taking more Cl⁻ with it ...
Eventually, an equilibrium is reached with:
Inside Outside
+
-
+
-
K80 40
Cl 20 40
A 60 0
total+80-80 +40-40
Positive and negative charges again balance on both sides of the membrane, so we still have space-charge neutrality.
The concentration gradient (in:out) for K⁺ is 80:40, and for Cl⁻ is 20:40, which gives a 2:1 ratio for both ions, and thus a Nernst potential of -17 mV for both ions. Everything balances!
Counting ions at the start, we find that positive and negative charges balance on each side of the membrane.
Inside Outside
+
-
+
-
K120
0
Cl60
0
A60
0
total+120-120 00
and thus we have space-charge neutralityThe principle of space-charge neutrality says that in a given volume the total positive charge is equal to the total negative charge. The only exception is very close to the cell membrane, where separation of charge gives rise to the membrane potential. However, the number of uncompensated ions needed to set up this potential is a very small fraction (<<1%) of the total available, so produces only a tiny deviation from the principle..
However, both K⁺ and Cl⁻ can cross the membrane ...
Explaining Donnan 3
and so both K⁺ and Cl⁻ flow down their concentration gradients out of the cell.
When 30 mM K⁺ and 30 mM Cl⁻ have moved to the outside of the cell, Cl⁻ is in chemical equilibrium with no gradient (30:30), but K⁺ still has a concentration gradient (90:30) driving it out of the cell ...
The principle of space-charge neutrality says that in a given volume the total positive charge is equal to the total negative charge. The only exception is very close to the cell membrane, where separation of charge gives rise to the membrane potential. However, the number of uncompensated ions needed to set up this potential is a very small fraction (<<1%) of the total available, so produces only a tiny deviation from the principle.
K 120
Cl 60
K 0
Cl 0
A 60
How Many Ions Move?
How do we know that 40 mM K⁺ and Cl⁻ had to move in the previous example? Consider the initial and final concentrations:
Initial
[K⁺in] = 120, [Cl⁻in] = 60
[K⁺out] = 0, [Cl⁻out] = 0
To maintain approximate space-charge neutrality, an almost equal amount [x] of K⁺ and Cl⁻ move across the membrane, therefore:
Final
[K⁺in] = 120 - x, [Cl⁻in] = 60 - x
[K⁺out] = x, [Cl⁻out] = x
The Donnan rule states that:
[K⁺in] x [Cl⁻in] = [K⁺out] x [Cl⁻out]
so at equilibrium:
(120 - x) x (60 - x) = x2
and thus
7200 - 180x +x2 = x2
So x = 40.
Conclusion
40 mM of KCl had to move across the membrane to set up the Donnan equilibrium.
Note: the final K⁺ and Cl⁻ ion distribution is independent of the starting distribution, so long as the ion totals are the same.
Squid giant axon
in out ENernst
[K⁺] 400 20 -75
[Cl⁻] 108 560 -41
[K⁺] x [Cl⁻] 4320011200
resting potential about -60 mV.
The squid axon does not obey the Donnan rule. The internal and external [K⁺] x [Cl⁻] products are very different, and the Nernst equilibrium potentials are different for K⁺ and Cl⁻, and different from the resting potential.
Is the Donnan Rule Obeyed?
Data source: Aidley, D.J. (1989) The physiology of excitable cells. 2nd edition, Cambridge University Press.
Frog and Squid
[Kin] x [Clin] = [Kout] x [Clout]
Frog muscle
in out ENernst
[K⁺] 124 2.25 -101
[Cl⁻] 1.5 77.5 -99
[K⁺] x [Cl⁻] 186174
resting potential -90 to -100 mV.
Frog muscle does approximately obey the Donnan rule. The internal and external [K⁺] x [Cl⁻] products are about equal, and the Nernst equilibrium potentials for both K⁺ and Cl⁻ are approximately equal to the resting potential.
Donnan equilibrium: Key points
1. The Donnan rule states that the product of the concentration of diffusible ions on one side of the membrane equals the product of the concentration of diffusible ions on the other.
2. The Donnan rule only applies to cells where the ions are passively distributed, i.e. there are no metabolic pumps using energy to regulate the ion concentration inside the cell.
3. The Donnan rule is important because, where it applies, it explains the origin of concentration gradients, and hence the resting potential.
4. The Donnan rule is approximately obeyed by frog muscle fibres (which were important preparations in early physiological experiments).
5. The Donnan rule is not obeyed by many nerve cells, where the resting potential is significantly different from the Nernst equilibrium potential of the major ions involved in setting it up.
What is happening when the Donnan rule is NOT obeyed?
Regulation is usually achieved by metabolic pumps such as the Na/K-ATPase (also known as the sodium-potassium exchange pump, or simply the sodium pump).
So far we have considered the situation where the membrane is permeable only to potassium and chloride ions, and their distribution is entirely passive.
However, most real nerve membranes are also somewhat permeable to sodium ions, which usually have a high extracellular concentration and a low intracellular concentration. Also, most nerves (and many other cells) regulateRegulation is usually achieved by metabolic pumps such as the Na/K-ATPase (also known as the sodium-potassium exchange pump, or simply the sodium pump).
the intracellular concentration of ions, so that they remain at a relatively fixed value.
We next consider what happens in this situation.
This will lead to an understanding of the Goldman-Hodgkin-Katz constant field equation, usually know as the Goldman equation for short.
The Goldman equation describes a steady-state condition, unlike the Nernst equation, which describes an equilibrium condition. The difference will be explained shortly ...
The Goldman Equation
Explaining Steady-State 1
Na+
gradient
Na
We start off with our familiar 2-compartment model with a 10:1 ratio of [K⁺] between the inside and outside compartments, but this time there is also a 1:10 ratio of [Na⁺] in the opposite direction.
The membrane will eventually be permeable to both K⁺ and Na⁺ ions, but to begin with, imagine the membrane is impermeable (all channels are shut). The membrane potential is therefore 0.
K
Explaining Steady-State 2
Now we open the K channels. Just as before, the membrane potential goes to the Nernst equilibrium potential for the K⁺ gradient, which is -58 mV.
We mark this as EK on the meter.
K Permeable
EK
Explaining Steady-State 3
Na Permeable
ENa
Next we close the K channels and open the Na channels. The membrane potential goes to the Nernst equilibrium potential for the Na⁺ gradient, which is +58 mV.
We mark this as ENa on the meter.
|
Explaining Steady-State 4
Now we open both sets of channels.
The membrane is permeable to K⁺, which tends to set the membrane potential to EK. However, the membrane is also permeable to Na⁺, which tends to set the membrane potential to ENa. There can only be one membrane potential, and it will be a “compromise” between the two equilibrium potentials.
With symmetrical gradients and equal permeability to both ions, the compromise potential is half way between the two equilibrium potentials, i.e. 0 mV.
What effect does this have on ion flow?
Na and K Permeable
Steady-State Fluxes
Explaining Steady-State 5
Since the membrane potential is not at either equilibrium potential, neither K⁺ nor Na⁺ are in equilibrium. Each has a steady flux down its concentration gradient.
(Note: In the absence of any restorative pumps, eventually the concentration gradients will run down, but this takes some time.)
Each ion experiences a driving force, which is the difference between its equilibrium potential (which itself would balance the concentration gradient) and the actual membrane potential.
driving force = Em - Eeq
The driving forces are equal and opposite, so the fluxesFlux in this case is the equivalent of current (charge per second). are equal and opposite. Neither side has a net change in charge, so there is no change in membrane potential.
Flux in this case is the equivalent of current (charge per second).
Unequal Na and K Permeability
Explaining Steady-State 6
Let us now suppose there are twice as many K channels as Na channels.
The membrane is thus highly permeable to K⁺, which tends to set the membrane potential to EK. However, the membrane is also somewhat permeable to Na⁺, which tends to drag the membrane potential a bit towards ENa.
The “compromise” potential is now between the two equilibrium potentials, but weighted towards the K end of the scale.
What effect does this have on the fluxes?
Each ion flux is driven by its driving force, which depends on how far the membrane potential is from the equilibrium potential.
There is thus a strong driving force on the Na⁺ ions, but only a weak driving force on the K⁺ ions.
On the other hand, the membrane permeability to K⁺ is high, but the membrane permeability to Na⁺ is low (note there are 2 K channels to 1 Na channel.
The result is that the Na⁺ flux (i.e. the number of Na⁺ ions crossing the membrane per second) is the same as the K⁺ flux. So again there is no change in the transmembrane charge distribution, and the membrane potential remains constant.
This is therefore a steady-stateIt is steady-state, but only in the short term. Eventually the gradients will run down, unless maintained by metabolic pumps.
That will be explained in more detail in a later section in the tutorial. condition.
It is steady-state, but only in the short term. Eventually the gradients will run down, unless maintained by metabolic pumps.
That will be explained in more detail in a later section in the tutorial.
Explaining Steady-State 7
Imbalance in Flux Self-Corrects
Explaining Steady-State 8
Why are the Na and K fluxes equal?
If more Na⁺ entered the inside compartment than K⁺ left it, then the inside compartment would become more postive. This would reduce the driving force on Na⁺ (because the membrane potential had moved closer to the Na equilibrium potential), and increase the driving force on K⁺.
The changes in driving force would reduce the inflow of Na⁺ ions and increase the outflow of K⁺ ions, thus counteracting the original imbalance.
VmV = -58 log
where α is the Na:K permeability ratio.
α
[Kin] + α [Nain]
I suggest you look at the pre-built examples first:
[Kout] + α [Naout]
Then try it out for yourself.
We have seen at a qualitative level that when the membrane is permeable to more than one ion species, the membrane potential is a weighted average of the Nernst equilibrium potentials for each permeant ion type.
This is quantified by the Goldman equation.
If we consider only Na and K, a simplified form of the Goldman equation is:
Simple Form of the Equation
With a completely symmetrical situation, the membrane potential is, not surprisingly, 0.
With 10:1 concentration ratios, the equilibrium potentials are +/- 58 mV.
200
Example 1 of 4
20
The Goldman Equation
1
Example 2 of 4
With Na⁺ permeability 5 times greater than K⁺ permeability (α = 5), the membrane potential moves towards the Na⁺ equilibrium potential, even though the concentration gradients are unchanged.
5
Increasing α obviously increases the influence of the Na⁺ gradient in the equation.
Example 3 of 4
From the equation it is clear that increasing the absolute Na⁺ concentrations by a factor of 5 has the same effect on the membrane potential as increasing α by a factor of 5.
The Na⁺ permeability is back equal to the K⁺ permeability, but now the absolute Na⁺ concentrations have increased 5 times. The membrane potential moves towards the Na⁺ equilibrium potential, even though the concentration gradients are unchanged.
1000
Example 4 of 4
Increasing the Na⁺ concentration gradient increases the Na⁺ equilibrium potential, and this has the effect of increasing the membrane potential.
Thus in summary, when the membrane is permeable to more than one ion, the membrane potential depends on:
the relative permeabilities, the absolute concentrations,the concentration gradients.
4. Relative Na⁺ : K⁺ permeability returns to its resting level, and so does the membrane potential.
+50 —
440
3. An increase in K⁺ permeability and a decrease Na⁺ permeability (α = 0.01) causes the membrane potential to go even more negative than the resting potential, towards the K⁺ equilibrium potential.
-50 —
400
Also known as a nerve impulse, or simply a spike.
0.04
1. In the resting state, the Na⁺ permeability is much lower than the K⁺ permeability (α = 0.04), and so the membrane potential is close to the K⁺ equilibrium potential.
mV 0 —
Action Potential
This is the preparation that Hodgkin and Huxley used in their classic study of the mechanism of the action potential, which won them a Nobel prize in 1963.
50
2. A massive increase in Na⁺ permeability (α = 20) causes the membrane potential to shift towards the Na⁺ equilibrium potential.
As a real example, let’s consider the squid giant axonThis is the preparation that Hodgkin and Huxley used in their classic study of the mechanism of the action potential, which won them a Nobel prize in 1963., which has approximately the concentration gradients shown here.
By varying the value of α (the permeability of Na relative to K), the membrane can generate the waveform of an action potentialAlso known as a nerve impulse, or simply a spike..
0.01
PK[Kin] + PNa[Nain] + PCl[Clout]
Cl⁻
0.45
where Pion is the membrane permeability to that ion.
The consequences of this depend on whether the neurons regulates the intracellular chloride concentration [Cl⁻in]. Some neurons do, some do not.
K⁺
1
So far we have ignored chloride ions in the Goldman equation. However, many neurons have a relatively high chloride permeability.
To take account of this, we need a more elaborate form of the Goldman equation:
Na⁺
0.04
Ion Permeability of the Resting Squid Giant Axon Relative to K
PK[Kout] + PNa[Naout] + PCl[Clin]
Membranes are Often Chloride Permeable
Regulated and Unregulated Chloride
Chloride Is Regulated
If intracellular Cl⁻ concentration is regulated, then it is unlikely that it will be exactly at the value necessary for Cl⁻ to be in Nernstian equilibrium. In this case the full version of the Goldman equation must be used to calculate the steady-state membrane potential.
Thus at steady-state, Cl⁻ is not in equilibrium, there is a flux of Cl⁻ across the membrane, and Cl⁻ ions do contribute to the resting membrane potential.
However, the regulated intracellular Cl⁻ concentration is usually such that the Cl⁻ equilibrium potential is quite close to the resting membrane potential.
Chloride Is Not Regulated
If intracellular Cl⁻ concentration is not regulated, then Cl⁻ ions will flow into or out of the neuron until the intracellular Cl⁻ concentration arrives at a value that has a Nernst equilibrium potential equal to the membrane potential set by the simple Goldman equation, which does not include chloride.
Thus at steady-state, Cl⁻ is in equilibrium across the membrane and does not contribute to the resting membrane potential.
The Goldman equation: Key points
1. Each ion has its own equilibrium (Nernst) potential, which is probably different from that of all the other ion types.
2. The actual membrane potential is a “compromise” between the various equilibrium potentials, each weighted by the membrane permeability and absolute concentration of the ion in question. This is described by the Goldman equation.
3. In the resting state potassium usually dominates the Goldman equation. Sodium permeability is low, so it only makes a small contribution. Chloride permeability is intermediate, but its equilibrium potential is usually quite close to the potassium equilibrium potential.
4. There will be a flux across the membrane of any ion whose equilibrium potential is not equal to the actual membrane potential.
5. If the membrane potential is stable, then the total net flow of ionic charge across the membrane is zero.
Leakage Channels
Not Much Known!
The hydrophobic phospholipid component of the cell membrane is virtually impermeable to charged particles such as ions. However, all membranes contain proteinaceous ion channels that allow certain ions to pass through. The most basic of these are called the leakage channels.
These form a sort of “background” permeability which is usually unchanging. Their absolute permeability is low compared to voltage- or ligand-gated channels (generating action and synaptic potentials respectively), but it is the leakage channels that are largely responsible for generating the resting potential.
?
Leakage channels are predominantly permeable to potassium, but have some sodium permeability as well. They mainly belong to a heterogenous collection of what are called tandem pore domainAlso sometimes called "2 pore domain" channels, these channels do not have two pores that cross the membrane. They have two pore loops in each segment of their transmembrane sub-unit structure. channels, and fall into classes with names such as TREK, TASK, TWIK or TALK
These channels do not have two pores that cross the membrane. They have two pore loops in each segment of their transmembrane sub-unit structure.
Gradient Run-Down
Run-Down Through Leakage?
The internal concentration now equals the external concentration.
This should not happen!
In the situation described by the Goldman equation, where the membrane is permeable to both K⁺ and Na⁺, the membrane potential is usually not at the equilibrium potential for either ion.
Therefore there is a continuous flux of both ions through the leakage channels, and, unless something else happened, the concentration gradients would run down.
Because the internal volume is small compared to the external volume, it would be the internal concentration that changed to become equal to the external concentration.
Also known as the Na-K ATPase, or simply the sodium pump.
What Stops Run-Down?
Concentration gradients are maintained by active ion pumps.
There are several different types of pump, but one of the most important is the Na-K exchangeAlso known as the Na/K-ATPase or simply the sodium pump. pump. This uses metabolic energyIn neurons, the pump accounts for up to 75% of energy expenditure. obtained by hydrolysing ATP to pump Na⁺ and K⁺ ions against their concentration gradients.
The pump has a transport ratio of 3:2, with 3 Na⁺ ions being pumped out of the cell for every 2 K⁺ ions pumped into it.
Na-K Exchange Pump 1
The Pump Balances the Passive Flux
Na
Na
Na
K
K
Na-K Exchange Pump 2
In the steady-state condition when the resting potential is stable, the 3 Na⁺ : 2 K⁺ ratio of the pump is exactly balanced by a 3 Na⁺ : 2 K⁺ flux ratio through the leakage channels.
If this were not so, and more positive ions continuously left the cell than entered it, the membrane potential would continuously increase.
(Note: it is the passive flux that adjusts to meet the pump ratio, not vice versa. The pump ratio is fixed by the molecular structure of the pump.
The pump pushes out more positive charge than it brings in, so it has a negative influence on the membrane potential.
The quantitative effect depends on the pump transport ratio r, which is 1.5 (3 Na⁺ : 2 K⁺). This can be expressed in a modified Goldman equation by increasing the influence of K⁺:
r [Kin] + α [Nain]
r [Kout] + α [Naout]
For the squid giant axon concentration and permeability values:
V = -64 mV (with pump: r = 1.5)
compared to
V = -60 mV (without pump: r = 1)
The Pump Contributes
to the Membrane Potential
The Pump is Electrogenic
The electrogenic Na-K exchange pump thus makes a contribution (usually just a few millivolts) to the resting membrane potential.
1. The Na-K exchange pump maintains the Na and K concentration gradients across the cell membrane. Without the pump, the gradients would inevitably run down due to flux through the leakage channels. (Note that the ion fluxes during action potentials and synaptic potentials would greatly increase the rate of run-down, and the pump rate has to increase in neurons with a high activity level to compensate.)
2. The pump hydrolyses ATP to provide the energy needed to move ions across the membrane against their concentration gradient.
3. The pump is electrogenic, and makes a direct contribution to the membrane potential, but it is usually not the dominant immediate cause of the membrane potential.
4. When the membrane potential is stable, the pump current is exactly balanced by the ion flux through the leakage channels.
The Na-K Exchange Pump: Key points
1. Neurons have high [K⁺in] and low [K⁺out], and a high membrane permeability to K⁺.
2. K⁺ tends to leave the neuron down its concentration gradient, producing an electrical gradient across the membrane with the inside negative relative to the outside.
3. The Nernst equation defines the equilibrium potential at which the electrical gradient balances the concentration gradient. Not many ions have to cross the membrane to produce equilibrium.
4. In a purely passive system, [K⁺] and [Cl⁻] gradients and a membrane potential can be set up by the Donnan equilibrium, if there are impermeant anions within the cell.
5. Most neurons have low [Na⁺in] and high [Na⁺out], and a low but significant resting permeability to Na⁺.
6. The actual resting membrane potential is a compromise between the negative equilibrium potential for K⁺ and and the positive equilibrium potential for Na⁺, weighted towards K because of its greater permeability. Cl⁻ plays a role if it is actively regulated, but not if it is passively distributed. The Goldman equation describes the resulting membrane potential.
7. Neither K⁺ nor Na⁺ are in equilibrium across the membrane. Cl⁻ may or may not be in equilibrium.
8. The passive flux of K⁺ and Na⁺ is balanced by active pumping by the Na-K exchange pump, so the pump is essential to maintain the concentration gradients. The pump is electrogenic and makes a contribution to the resting membrane potential, but this is usually quite small.
9. The main immediate cause of the resting membrane potential in most neurons is the differential distribution of sodium and potassium ions across the membrane, and the different permeability of the membrane to those ions.
Overall Summary
The Origin of
the
Resting Membrane Potential
Author's
home page
Dr W. J. Heitler
School of Psychology and Neuroscience
University of St Andrews
Scotland
All living cells have a potential difference (voltage) across their cell membranes, with the inside usually negative relative to the outside. In nerve cells the value of the resting potential varies between about -40 and -90 mV.
The voltage can be measured by penetrating the cell with a microelectrodeA microelectrode is a specialized electrode for recording voltages from the inside of cells. It typically consists of a glass tube drawn out to a very fine point and filled with an electrolyte solution such as potassium chloride. The sharp tip of the electrode penetrates through the cell membrane. The blunt end of the electrode is connected by a wire to a voltage measuring instrument.. This is connected to a specialized voltmeterInstruments for recording voltages from cells have to have a very high input impedance. This means that they can measure the voltage without drawing much current from the cell, and thus without changing the very voltage that they are trying to measure. which measures the potential difference between the inside of the cell and the surrounding environment. This is the membrane potential.
This tutorial describes the cellular mechanisms that generate the resting membrane potential.
Early in the 20th century, Bernstein proposed that the resting membrane potential was due to 3 factors:
the cell membrane is selectively permeable to potassium ions,the intracellular potassium concentration is high,the extracellular potassium concentration is low.This is known as the potassium electrode hypothesis.
In the next section we will show how these factors would cause a membrane potential with the inside negative.
This will lead to an understanding of the Nernst equation and the ionic equilibrium (reversal) potentials.
Sneak preview - Bernstein was almost, but not completely, right.
In our scenario there is initially a large driving force (shown as an arrow through the pore). This is because the chemical gradient is not balanced by any electrical gradient.
The difference between the chemical gradient and the electrical gradient is called the driving force.
and rearranging ...
3. Gives the Nernst equation.
Deriving the Nernst equation 2
1. At equilibrium the energies are equal, so the two equations are equal.
2. Losing common factors n and δW
Deriving the Nernst equation 3
K⁺ Ca⁺⁺ Cl⁻
If we reverse the direction of the concentration gradient, so that the ratio is now 1:10 (i.e. 0.1), then the absolute value of the membrane potential stays the same, but its polarity reversesNote that the ion has changed to sodium but the Nernst equation doesn’t care - it is still a monovalent ion. (We assume the membrane is now permeable to sodium.) (the log of 0.1 is -1).
1. The Nernst equation defines the electrical potential across a membrane that will balance a particular chemical concentration gradient of an ion. So if the membrane potential is at the Nernst value, there will be no net movement of that ion across the membrane, even if the membrane is permeable to the ion.
2. This potential is called the equilibrium or reversal potential for that ion. It depends on the concentration gradient (ratio) and the valency of the ion (and the temperature), but does not depend on the degree of permeability of the membrane to the ion, nor on the absolute concentrations.
3. Each type of ion will have its own equilibrium potential, and this is likely to be different to that of other ion types.
4. If a membrane is permeable to only ONE type of ion, then the membrane potential will automatically move to the equilibrium potential for that ion.
5. Relatively few ions have to move to set up the potential, so the concentration gradient is not significantly perturbed in achieving equilibrium.
However, what if the cell membrane is permeable to more than one type of ion …?
Many cells have membranes with quite a high permeability to both potassium and chloride ions. In the absence of anything else, this would prevent the establishment of the membrane potential, because Cl⁻ would simply move down the concentration gradient along with the K⁺, and counteract the positive charge. The concentrations would just equilibrate across the membrane.
However, most cells also have a relatively high intracellular concentration of large molecular weight anions A⁻ (e.g. negatively charged proteins).
The intracellular composition is thus:
K⁺Cl⁻ + K⁺A⁻
This can result in a Donnan equilibrium, which both maintains opposite K⁺ and Cl⁻ gradients across the membrane, and sets up a resting potential.
The Donnan equilibrium applies to ions whose distribution is passive and unregulated, i.e. there are no metabolic pumps maintaining the intracellular concentrations at fixed values.
Now consider the situation where some intracellular Cl⁻ ions are replaced by the impermeant anion A⁻. The K⁺ and Cl⁻ ions can still flow down their concentration gradients and leave the cell, but A⁻ cannot pass through the membrane.
In the diagram each symbol represents 10 mM of substance - e.g. there are 12 K⁺s giving a total of 120 mM.
We start off with everything inside the cell, nothing outside.
What happens now?
Eventually, an equilibrium is reached with:
Inside Outside
+
-
+
-
K80 40
Cl 20 40
A 60 0
total+80-80 +40-40
We still have space-charge neutrality, and the concentration gradient (in:out) for K⁺ is 80:40, and for Cl⁻ is 20:40, which gives a 2:1 ratio for both ions, and thus a Nernst potential of -17 mV for both ions. Everything balances!
so at equilibrium:
(120 - x) x (60 - x) = x2
and thus
7200 - 180x +x2 = x2
So x = 40.
Conclusion
40 mM of KCl had to move across the membrane to set up the Donnan equilibrium.
Note: the final K⁺ and Cl⁻ ion distribution is independent of the starting distribution, so long as the ion totals are the same.
1. The Donnan rule states that the product of the concentration of diffusible ions on one side of the membrane equals the product of the concentration of diffusible ions on the other.
2. The Donnan rule only applies to cells where the ions are passively distributed, i.e. there are no metabolic pumps using energy to regulate the ion concentration inside the cell.
3. The Donnan rule is important because, where it applies, it explains the origin of concentration gradients, and hence the resting potential.
4. The Donnan rule is approximately obeyed by frog muscle fibres (which were important preparations in early physiological experiments).
5. The Donnan rule is not obeyed by many nerve cells, where the resting potential is significantly different from the Nernst equilibrium potential of the major ions involved in setting it up.
What is happening when the Donnan rule is NOT obeyed?
Donnan equilibrium: Key points
So far we have considered the situation where the membrane is permeable only to potassium and chloride ions, and their distribution is entirely passive.
However, most real nerve membranes are also somewhat permeable to sodium ions, which usually have a high extracellular concentration and a low intracellular concentration. Also, most nerves (and many other cells) regulateRegulation is usually achieved by metabolic pumps such as the Na/K-ATPase (also known as the sodium-potassium exchange pump, or simply the sodium pump).
the intracellular concentration of ions, so that they remain at a relatively fixed value.
We next consider what happens in this situation.
This will lead to an understanding of the Goldman-Hodgkin-Katz constant field equation, usually know as the Goldman equation for short.
The Goldman equation describes a steady-state condition, unlike the Nernst equation, which describes an equilibrium condition. The difference will be explained shortly ...
Since the membrane potential is not at either equilibrium potential, neither K⁺ nor Na⁺ are in equilibrium. Each has a steady flux down its concentration gradient.
(Note: In the absence of any restorative pumps, eventually the concentration gradients will run down, but this takes some time.)
Each ion experiences a driving force, which is the difference between its equilibrium potential (which itself would balance the concentration gradient) and the actual membrane potential.
driving force = Em - Eeq
The driving forces are equal and opposite, so the fluxesFlux in this case is the equivalent of current (charge per second). are equal and opposite. Neither side has a net change in charge, so there is no change in membrane potential.
previous
1. Each ion has its own equilibrium (Nernst) potential, which is probably different from that of all the other ion types.
2. The actual membrane potential is a “compromise” between the various equilibrium potentials, each weighted by the membrane permeability and absolute concentration of the ion in question. This is described by the Goldman equation.
3. In the resting state potassium usually dominates the Goldman equation. Sodium permeability is low, so it only makes a small contribution. Chloride permeability is intermediate, but its equilibrium potential is usually quite close to the potassium equilibrium potential.
4. There will be a flux across the membrane of any ion whose equilibrium potential is not equal to the actual membrane potential.
5. If the membrane potential is stable, then the total net flow of ionic charge across the membrane is zero.
For the squid giant axon concentration and permeability values:
V = -64 mV (with pump: r = 1.5)
V = -60 mV (without pump: r = 1)
Overall Summary
In most neurons the electrogenic Na/K pump makes only a minor contribution to the resting potential. However, the actual amount is variable, and in some specialised neurons it may be considerable.
One unusal example occurs in olfactory receptor neurons in frogs (and proably mammals too). These have an extremely high membrane resistance, which makes them very sensitive because a small stimulus current produces a large voltage response.
The high membrane resistance is because they have virtually no resting potassium conductance. So the Bernstein hypothesis is completely wrong for these neurons!
Instead, in these neurons the resting potential is almost entirely due to the electrogenic Na/K pump.
The pump on its own would produce a membrane potential of about -140 mV. However, the pump-induced hyperpolarization activates a mixed cation channel which produces a depolarizing current (Ih, sometimes called the “funny current” because it is unusual in being activated by hyperpolarization) that works against the pump current.
The end result is a resting potential of about -80 mV, which is within the normal range. But it is produced by a very abnormal mechanism!
Trotier & Døving (1996) Functional role of receptor neurons in encoding olfactory information. J. Neurobiol. 30; 58-66.
Unusual Mechanisms
In most neurons the electrogenic Na/K pump makes only a minor direct contribution to the resting potential. However, the actual amount is variable, and in some specialised neurons it may be considerable.
One such example occurs in olfactory receptor neurons in frogs (and probably mammals too). These have an extremely high membrane resistance, which makes them very sensitive because a small stimulus current produces a large voltage response.
The high membrane resistance is because they have virtually no resting potassium conductance. So the Bernstein hypothesis is completely wrong for these neurons!
Instead, in these neurons the resting potential is almost entirely due to the electrogenic Na/K pump.
The pump on its own would produce a membrane potential of about -140 mV. However, the pump-induced hyperpolarization activates a mixed cation channel which produces a depolarizing current (Ih, sometimes called the “funny current” because it is unusual in being activated by hyperpolarization) that works against the pump current.
The end result is a resting potential of about -80 mV, which is within the normal range. But it is produced by a very abnormal mechanism!