## Goldmanhodgkinkatz Equation

The Goldman equation is also known as the Goldman-Hodgkin-Katz (GHK) equation because Hodgkin and Katz applied it to biologic membranes. As has already been seen in Chapter 3, the GHK equation can be used to determine the potential developed across a membrane permeable to Na+ and K+. Thus,

where Vm is the membrane potential in millivolts and a is equal to the ratio of the Na+ and K+ permeabilities (PNa/PK). This equation looks rather complex at first, but it can be simplified by examining two extreme cases. Consider the case when the Na+ permeability is equal to zero. Then, a is equal to zero, and the Goldman-Hodgkin-Katz equation reduces to the Nernst equation for K+. If the membrane is highly permeable to Na+ and has a very low K+ permeability, a will be a very large number, which causes the Na+ terms to be very large so that the K+ terms can be neglected and the Goldman-Hodgkin-Katz equation reduces to the Nernst equation for Na+. Thus, the Goldman-Hodgkin-Katz equation has two extremes. In one case, when Na+ permeability is zero, it reduces to the Nernst equation for K+; in the other case, when Na+ permeability is very high, it reduces to the Nernst equation for Na+. The GHK equation allows one to predict membrane potentials between these two extreme levels, and these membrane potentials are determined by the ratio of K+ and Na+ permeabilities. If the permeabilities are equal, the membrane potential will be intermediate between the K+ and the Na+ equilibrium potentials.

Figure 5 illustrates a test of the ability of the Goldman-Hodgkin-Katz equation to fit the same experimental data shown in Fig. 4. The straight line is generated by the Nernst equation, whereas the curved trace is generated by the Goldman-Hodgkin-Katz equation. The value of a that gives the best fit is 0.01. Thus, although there is some Na+ permeability at rest, it is only one-hundredth that of the K+ permeability. To a first approximation, the membrane potential is due to the fact that there is unequal distribution of K+, and the membrane is selectively permeable to K+ and to a large extent no other ion. Therefore, the membrane potential can be roughly predicted by the Nernst equilibrium potentials for K+. However, there is a slight Na+ permeability that tends to make the inside of the cell more positive than would be predicted, based on the assumption that the cell is permeable only to K+. The GHK equation can be used to calculate or predict

o CL

0 0.2 0.5 1.0 2.5 5 10 20 50 100 Potassium Concentration (mM)

FIGURE 5 Same experiment as Fig. 4, but the graph also contains the prediction of the change in membrane potential obtained with the Goldman-Hodgkin-Katz equation with a value of a equal to 0.01. (Modified from Hodgkin AL, Horowicz P. J Physiol 1959; 148:127.)

0 0.2 0.5 1.0 2.5 5 10 20 50 100 Potassium Concentration (mM)

FIGURE 5 Same experiment as Fig. 4, but the graph also contains the prediction of the change in membrane potential obtained with the Goldman-Hodgkin-Katz equation with a value of a equal to 0.01. (Modified from Hodgkin AL, Horowicz P. J Physiol 1959; 148:127.)

the membrane potential knowing the ratio of Na+ and K+ permeabilities and the individual extracellular and intracellular concentrations of Na+ and K+.

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