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) }](/images/math/c/1/b/c1b5c1cf3dbccab5482625658b15f1c2.png)
This results in the following if we consider a membrane separating two KxNa1 − xCl-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) }](/images/math/8/5/1/851aa82025472a62b619ccb167f5c0a6.png)
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) }](/images/math/8/4/3/843f1ab5f23a901fd66fe7a27b4a3f27.png)
- Em = The membrane potential
- Pion = 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) }](/images/math/b/f/4/bf4a0a5cc0db1ed0a9e9208fb80017cd.png)
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. 
). 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. 
).
See also
External links
- Nernst/Goldman Equation Simulator
- Nernst/Goldman interactive Java applet The membrane voltage is calculated interactively as the number of ions are changed between the inside and outside of the cell.
Acknowledgement and Attribution Regarding Sources of Content
Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .
