The Goldman Equation

The examples discussed so far have been concerned with the qualitative relation between membrane potential and relative ionic permeabilities. The equation that gives the quantitative relation between Em on the one hand and ion concentrations and permeabilities on the other is the Goldman equation, which is also called the constant-field equation. For a cell that is permeable to potassium, sodium, and chloride, the Goldman equation can be written as:

This equation is similar to the Nernst equation (see Chapter 4), except that it simultaneously takes into account the contributions of all permeant ions. Some information about the derivation of the Goldman equation can be found in Appendix B. Note that the concentration of each ion on the right side of the equation is scaled according to its permeability, p. Thus, if the cell is highly permeable to potassium, for example, the potassium term on the right will dominate and Em will be near the Nernst potential for potassium. Note also that if pNa and pcl were zero, the Goldman equation would reduce to the Nernst equation for potassium, and Em would be exactly equal to EK, as we would expect if the only permeant ion were potassium.

Because it is easier to measure relative ion permeabilities than it is to measure absolute permeabilities, the Goldman equation is often written in a slightly different form:

In this case, the permeabilities have been expressed relative to the permeability of the membrane to potassium. Thus, b = pNa/pK, and c = pCl/pK. We have also evaluated RT/F at room temperature, converted from ln to log, and expressed the result in millivolts.

For most nerve cells, the Goldman equation can be simplified even further: the chloride term on the right can be dropped altogether. This approximation is valid because the contribution of chloride to the resting membrane potential is insignificant in most nerve cells. In this case, the Goldman equation becomes

This is the form typically encountered in neurophysiology. In nerve cells, the ratio of sodium to potassium permeability, b, is commonly about 0.02,

Figure 5-4 Experimentally determined relation between external potassium concentration and resting membrane potential of an axon in the spinal cord of the lamprey. The circles show the measured value of membrane potential at five different values of [K+]o. The dashed line gives the potassium equilibrium potential calculated from the Nernst equation. The solid line shows the prediction from the Goldman equation with internal and external sodium and potassium concentrations appropriate for the lamprey nervous system.

although this value may vary somewhat from one type of cell to another. That is, pK is about 50 times higher than pNa. Thus, Equation (5-3) tells us that Em would be about -71 mV for a cell with [K+] = 125 mM, [K+]o = 5 mM, [Na+] = 12 mM, [Na+]o = 120 mM, and b = 0.02. What would Embe for the same cell if b were 1.0 (that is, if pNa = pK) instead of 0.02?

The Goldman equation tells us quantitatively what we would expect qualitatively. If pK is 50 times higher that pNa, we would expect Em to be nearer to EK than to ENa. Indeed, Equation (5-3) yields Em = -71 mV, which is much nearer to Ek (-80 mV) than to ENa (+58 mV). The difference between Em and EK reflects the steady influx of sodium ions carrying positive charge into the cell and maintaining a depolarization from EK.

The applicability of the Goldman equation to a real cell can be tested experimentally by varying the concentration of potassium in the ECF and measuring the resulting changes in membrane potential. If membrane potential were determined solely by the distribution of potassium ions across the cell membrane that is, if the factor b in Equation (5-3) were zero we know that Em would be determined by the potassium equilibrium potential. In this situation, a plot of measured membrane potential against log [K+]o would yield a straight line with a slope of 58 mV per tenfold change in [K+]o. This straight line would merely be a plot of the Em calculated from the Nernst equation at different values for external potassium concentration, and it is shown by the dashed line in Figure 5-4. Look, however, at the actual data from a real experiment in Figure 5-4. These data show the measured values of Em of a nerve fiber observed at a number of different external potassium concentrations. The data do not follow the line expected from the Nernst equation, but instead fall along the solid line. That line was drawn according to the form of the Goldman equation given in equation (5-3), and this experiment demonstrates that the real value of membrane potential in the nerve fiber is determined jointly by potassium and sodium ions. Experiments of this type by Hodgkin and Katz in 1949 first demonstrated the role of sodium ions in the resting membrane potential of real cells.

Equation (5-3) is a reasonable approximation to Equation (5-2) only if pCl/pK is negligible. To determine if it is valid to ignore the contribution of chloride that is, to use Equation (5-3) experiments like that summarized in Figure 5-4 can be performed in which the concentration of chloride in the ECF is varied rather than the concentration of potassium. When that was done on the type of nerve cell used in the experiment of Figure 5-4, it was found that a tenfold reduction of [Cl-]o caused only a 2 mV change in the resting membrane potential. Thus, for that type of cell, membrane potential is relatively unaffected by chloride concentration, and Equation (5-3) is valid. This is also true for other nerve cells. It is important to emphasize, however, that the membranes of other kinds of cells, such as muscle cells, have larger chloride permeability; therefore, the membrane potential ofthose cells would be more strongly dependent on external chloride concentration. This has been demonstrated experimentally for muscle cells by Hodgkin and Horowicz.

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