5.6: Calculating Ex and the Nernst equation

Even though the equilibrium potential (E_x) for each ion can be determined experimentally, it is also possible to calculate E_x using a mathematical equation called the Nernst equation. By understanding the Nernst equation, you can predict the direction that ions will move when an ion channel opens given various conditions. The Nernst equation is as follows:

\[E_x=\frac{R T}{z F} \ln \left(\frac{[x]_o}{[x]_i}\right)\nonumber\]

Equation 4.9 The Nernst equation calculates the reversal potential for a given ion x.

E_x is the equilibrium potential. As a potential, it has units in volts, usually measured in millivolts. When a cell’s membrane potential is exactly at this E_x, the ion is at equilibrium: ion movement into the cell is matched by ion movement out of the cell. E_x is sometimes also called the reversal potential, because once the cell membrane crosses from this potential, the net movement of the ions reverses directions. This value is also called the Nernst potential in honor of Walther Nernst, the Nobel prize-winning German chemist who first developed the formula to describe electrochemical reactions.

R is the ideal gas constant. It has a value of 8.314 J/K*mol. The R term is included in the equation because it is a way to convert between the number of molecules and the energy that these molecules exert.

T is the temperature in Kelvin (to convert from Celsius into Kelvin, add 273 to the Celsius value. C + 273 = K). Usually, room temperature is around 293 K, and biological temperatures are closer to 310 K. This term is included in the equation since all biological processes are temperature dependent, especially those at the level of molecular machinery.

z is the electrical charge of the ion. For sodium and potassium, z is +1, for chloride or monovalent anions z is -1, and for divalent cations like calcium and magnesium, z is +2. This term is essential since the Nernst potential is used to predict the direction of ion movement based on the charge of the ion. This value represents the influence exerted by the electrical gradient.

F is the Faraday constant which has a value of 96,485 Coulombs / mol.

[x]_o and [x]_i represent the concentration of ion x outside the cell and inside the cell, respectively. Their units are generally in mM, but these units cancel out in the equation. These values represent the forces acting on the ions by the chemical gradient.

The Nernst equation has many complex constants and terms, so neuroscientists often use the back-of-the-envelope equation as a shortcut for quickly calculating the equilibrium potential. This shortcut equation condenses down the R and F constants, turns the natural log into a base 10 logarithm, and assumes the calculations are done at physiological temperature. The shortcut formula can be written as:

\[E_x \cong \frac{61}{z} \log \left(\frac{[x]_o}{[x]_i}\right)\nonumber\]

Equation 4.10 The “back-of-the-envelope” equation is a shortcut to estimate the reversal potential for an ion x.

The Nernst equation is able to calculate the reversal potential for individual ions assuming that the appropriate ion channels are open. However, in neurons, not all ion channels are opened or closed at the same time. In a neuron at rest, usually Na+ does not enter into the cells since sodium channels are closed. K+ ions , on the other hand, are often moving through leak channels all the time. Chloride channels are generally open as well, but they pass less current at rest than potassium channels do.

A formula called the Goldman-HodgkinKatz equation (GHK equation) combines the Nernst potentials of three relevant ions (Na⁺, K⁺, and Cl^-) into a single equation that, when evaluated, gives us the value of the membrane potential V_m.

The GHK equation is written as:

\[V_m=\frac{R T}{F} \ln \frac{p_K\left[\mathrm{~K}^{+}\right]_o+p_{\mathrm{Na}}\left[\mathrm{Na}^{+}\right]_o+p_{\mathrm{Cl}}\left[\mathrm{Cl}^{-}\right]_i}{p_K\left[\mathrm{~K}^{+}\right]_i+p_{\mathrm{Na}}\left[\mathrm{Na}^{+}\right]_i+p_{\mathrm{Cl}}\left[\mathrm{Cl}^{-}\right]_o}\nonumber\]

Equation 4.11 The Goldman-Hodgkin-Katz equation is used to calculate the membrane potential given permeability of ions and their concentrations across the cell membrane.

When you analyze the GHK equation, you will notice that it is essentially a combination of the Nernst equations for the equilibrium potentials for the three ions. The GHK equation also introduces a new term, the value p which stands for permeability: the ability for an ion to cross the membrane through ion channels. Permeability itself does not have a unit.

It is easier to think of permeability as the “weight” of each equilibrium potential. The higher the permeability for a given ion, the closer the V_m is to the E_x for that ion. For example, consider the value of the GHK equation for neurons at rest. Under these conditions, the permeability for K⁺ (p_K) is 1, p_Cl is 0.55, and p_Na is 0.04. Therefore, the resting membrane potential V_m will be closest to a combination of E_K and E_Cl, since these two terms dominate the GHK equation. Just as a reminder, the resting membrane potential is around -70 mV, and E_K is around -80 mV while E_Cl is around -60 mV.

However, during an action potential (described in section 4.4), permeability for sodium increases significantly. As pNa rises, the membrane potential will shift closer towards E_Na, which is +55 mV. The GHK equation provides a mathematical explanation for how movement of Na+ across the membrane causes the cell to become more positive.

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