# Reversal potential

The reversal potential (also known as the Nernst potential) of an ion is the potential of the cell membrane at which there is no net (overall) flow of ions from one side of the membrane to the other. It defines the negative resting potential of neurons, muscle cells and other excitable cells when the cell is quite (not exited). In these cells the reversal potential is created by the selective membrane permeability to K+.

Diffusion of uncharged particles, diffusion of ions, diffusion with selective membrane present and raise of the reversal potential. In this example, the + 50 mV reversal potential at the end compensates the -50 mV external potential.

In a single-ion system, reversal potential is also the equilibrium potential (numerical values are identical). "Equilibrium" means that at this voltage, both outward and inward rates of ion movement are the same; the ion flux is in equilibrium. Ions still may move but the net current is zero. "Reversal" means that a change of membrane potential on either side of the equilibrium potential reverses the overall direction of ion flux. However multi-ion systems may be in the state when the summed currents of the multiple ions equals zero. While this is a reversal potential in the sense that membrane current reverses direction, it is not an equilibrium potential because some (frequently all) of the ions are not in equilibrium and thus have net fluxes across the membrane. When a cell has significant permeabilities to more than one ion, Nernst equation is not suitable and the Goldman-Hodgkin-Katz equation is required to calculate the potential.

The fact that the reversal potential for a particular membrane matches the equilibrium potential for a particular ion is the experimental proof that this membrane contains channels that are specific for that ion[1].

Membrane itself has little permeability for charged particles. Hence most of the ion flow across is is caused by the presence a specialized proteins, ion channels. These channels are often highly specific to one type of ion (K+, Na+, etc). When a channel type that is selective to one species of ion dominates (because other ion channels are closed, for example) then the voltage inside the cell is will equilibrate (i.e. become equal) to the reversal potential for that ion (assuming 0 outside the cell). For the most cells, potassium conductance dominates during the resting stage and so the resting potential is close to the K+ (potassium ion) reversal potential. During a typical action potential, the large number of Na+ channels open, bringing the membrane potential close to the reversal potential of Na+.

## Nernst equation

The reversal potential can be calculated from the Nernst equation (so it is also called Nernst potential).

The term driving force is related to equilibrium potential, and is likewise useful in understanding the current in biological membranes. Driving force refers to the difference between an ion's equilibrium potential and the actual membrane potential. It is defined by the following equation:

[[Math:c|{Driving Force} = ({V_m}-{E_{ion}})\,]]
[[Math:c|{I_{ion}} = {G_{ion}} ({Driving Force})\,]]
[[Math:c|{I_{ion}} = {G_{ion}} ({V_m}-{E_{ion}})\,]]

In words, this equation says that: the ionic current [[Math:c|{I_{ion}}]] is equal to that ion's conductance [[Math:c|{G_{ion}}]] multiplied by the driving force, the difference between the membrane potential and the ion's equilibrium potential [[Math:c|{V_m}-{E_{ion}}]]. The ionic current will always be zero if the membrane is impermeable ([[Math:c|{G_{ion}} = 0]]) to the ion in question.

## Finding the Reversal potential

Probability that a ion takes a state of energy E is proportional to the Boltzmann factor

[[Math:c| p(E) \sim exp \bigg( { - E \over { k \cdot T}} \bigg) ]]

where T the temperature and k is the Boltzmann constant. This energy at location x is equal to

[[Math:c| E(x) = q \cdot u_x ]]

where [[Math:c|u_x]] is the potential at the location x. Hence the probability to find a ion somewhere around x is proportional to

[[Math:c| exp \bigg (- q \cdot {u_x \over {k \cdot T}} \bigg). ]]

where q is the charge of the ion. The number of ions is sufficiently huge to interpret probability as the actual density. Now let's assume that the position x1 and x2 are on the opposite surfaces of the membrane right across the lipid layer. For the positively charged ions, the ion density ration between the point x1 and x2 is

[[Math:c| {n_{x1}} \over {n_{x2}} = exp \bigg ( - {{ q \cdot ({u_{x1}} - {u_{x2}}) } \over {k \cdot T}} \bigg) ]]

Here

[[Math:c| {u_{x1}} - {u_{x2}} = \Delta u ]]

is the difference of electric potentials over two positions that is required to find. It can be found from the expressions above as

[[Math:c| \Delta u = {k \cdot T} \over q ln { {n_2} \over {n_1} } ]][2].

where n1 and n1 are the molar concentrations of the ion on both sides of the membrane. For the resting cell with known usual inner (about 400 mM/l) and outer (about 20 mM/l) concentrations of K+ the resting potential is about -77. As the higher K+ concentration is inside, the cell is negatively charged. The computed potential does not depend on the membrane permeability (as long as it is non zero). The formula assumes that the volumes inside and outside are big enough not to impact the concentrations during formation of the potential (close to the truth for the most of living cells).

Some sources referenced in the literature list express this equation through Avogadro constant [[Math:c|N_A]] (the number of ions in one moll), Faraday constant ([[Math:c|F = N_A \cdot q]] for monovalent ions) and universal gas constant ([[Math:c|R = k \cdot N_A]]). From here,

[[Math:c|q = F \over {N_A} \to \Delta u = {k \cdot N_A \cdot T} \over F ln { {n_2} \over {n_1} } = {R \cdot T} \over F ln { {n_2} \over {n_1} }]]

## Coupled transport

Some ion transporters transfer multiple ions and uncharged molecules during they operating cycle. For instance, a sodium pump may transfer 3 K+ ions inside and 2 Na+ ions outside the cell. If the total electric charge of all transferred ions remains non zero, a membrane where such transporter is dominant still have the reversal potential. When transporter transfers x Na+ ions outside in exchange of y K+ ions inside and uses no additional energy, the formula of the reversal potential can be derived from (B. Chapman 1978)[3] as

[[Math:c|\Delta u = {RT} \over {(x-y)F} \cdot \Bigg ( x \cdot ln {{Na}_{in}} \over {{Na}_{out}} \plus y \cdot ln {K_{out}} \over {K_{in}} \Bigg )]]

From this formula it is obvious that reversal potential is only defined when [[Math:c|x \ne y]]. Transferring the equal number ions of the same valence in the opposite directions has no impact on existing reversal potential, if any.

Transporter can also couple transport of the charged ions with the transport of neutral molecules. In such case the chemical gradient of these "passenger molecules" also has impact on reversal potential. For instance, GABA transporter, described in [4]) co-translocates a neutral molecule of γ-aminobutyric acid (GABA), two Na+ ions and one Cl- ion across the plasma membrane during a single cycle of operation. The reversal potential then can be computed from formula

[[Math:c|\Delta u = {RT} \over {(2-1)F} \cdot ln \Bigg ( {{GABA}_{in}} \over {{GABA}_{out}} \plus \Bigg ( {{Na}_{in}} \over {{Na}_{out}} \Bigg )^2 \plus {{Cl}_{in}} \over {{Cl}_{out}} \Bigg )]]

Here 2-1 means

[[Math:c|2 \cdot z_{Na} - z_{Cl}]],

where [[Math:c|z_{Na}]] is the valence of Na+ and [[Math:c|z_{Cl}]] is the valence of Cl- (both equal to one).

## Active transport

Reversal potential can also be computed for the membrane that contains active ion transporters. Such transporters convert between chemical energy (usually ATP) and the energy of the ion electrochemical gradient, but they convert in both directions. At the reversal potential, not just the ion currents are balanced but also the ATP synthesis rate is equal to the ATP breakdown rate. The calculated potential depends on the free energy of the ATP breakdown. It also depends on how many ions are transferred in both directions while breaking or synthesizing the single ATP molecule in exchange. B. Chapman (1978) extends the formula for K+/Na+ antiporter for the usual case when this transporter also breaks an ATP molecule during its cycle:

[[Math:c|\Delta u = {RT} \over {(x-y)F} \cdot \Bigg ( x \cdot ln {{Na}_{in}} \over {{Na}_{out}} \plus y \cdot ln {K_{out}} \over {K_{in}} - A / RT \Bigg )]]

where A is a free energy of the ATP breakdown. The formula has also been used to determine this energy under physiological conditions.

## References

1. 1 R.Norman (2003) Reversal potential/Equilibrium potential Definitions!
2. 2 Lecture node at icwww.epfl.ch
3. 3 B. Chapman (1978) The reversal potential] of the electrogenic sodium pump
4. 4 Lu & Hilgemann (1999); Richerson & Wu, (2003) GABA Transporter Reversal Potential