# Goldman equation

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## Overview

The Goldman-Hodgkin-Katz voltage equation, more commonly known as the Goldman equation is used in cell membrane physiology to determine the potential across a cell's membrane taking into account all of the ions that are permeant through that membrane.

The discoverers of this are David E. Goldman of Columbia University, and the English Nobel laureates Alan Lloyd Hodgkin and Bernard Katz.

## The equation

The GHK voltage equation for $N$ monovalent positive ionic species and $M$ negative:

$E_{m} = \frac{RT}{F} \ln{ \left( \frac{ \sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_{out} + \sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_{in}}{ \sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_{in} + \sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_{out}} \right) }$

This results in the following if we consider a membrane separating two $K_{x}Na_{1-x}Cl$-solutions:

$E_{m, K_{x}Na_{1-x}Cl } = \frac{RT}{F} \ln{ \left( \frac{ P_{Na^{+}}[Na^{+}]_{out} + P_{K^{+}}[K^{+}]_{out} + P_{Cl^{-}}[Cl^{-}]_{in} }{ P_{Na^{+}}[Na^{+}]_{in} + P_{K^{+}}[K^{+}]_{in} + P_{Cl^{-}}[Cl^{-}]_{out} } \right) }$

It is "Nernst-like" but has a term for each permeant ion. The Nernst equation can be considered a special case of the Goldman equation for only one ion:

$E_{m,Na} = \frac{RT}{F} \ln{ \left( \frac{ P_{Na^{+}}[Na^{+}]_{out}}{ P_{Na^{+}}[Na^{+}]_{in}} \right) }=\frac{RT}{F} \ln{ \left( \frac{ [Na^{+}]_{out}}{ [Na^{+}]_{in}} \right) }$
• $E_{m}$ = The membrane potential
• $P_{ion}$ = the permeability for that ion
• $[ion]_{out}$ = the extracellular concentration of that ion
• $[ion]_{in}$ = the intracellular concentration of that ion
• $R$ = The ideal gas constant
• $T$ = The temperature in kelvins
• $F$ = Faraday's constant

The first term, before the parenthesis, can be reduced to 61.5 log for calculations at human body temperature (37 C)

$E_{X} = 61.5 \log{ \left( \frac{ [X^{+}]_{out}}{ [X^{+}]_{in}} \right) } = -61.5 \log{ \left( \frac{ [X^{-}]_{out}}{ [X^{-}]_{in}} \right) }$

Note that the ionic charge determines the sign of the membrane potential contribution.

The usefulness of the GHK equation to electrophysiologists is that it allows one to calculate the predicted membrane potential for any set of specified permeabilities. For example, if one wanted to calculate the resting potential of a cell, they would use the values of ion permeability that are present at rest (e.g. $P_{K^{+}} >> P_{Na^{+}}$). If one wanted to calculate the peak voltage of an action potential, one would simply substitute the permeabilities that are present at that time (e.g. $P_{Na^{+}} >> P_{K^{+}}$).