Membrane Potentials and Action Potentials

The Nernst Equation Describes the Relationship of Diffusion Potential to the Ion Concentration Difference Across a Membrane

The diffusion potential across a membrane that exactly opposes the net diffusion of a particular ion through the membrane is called the Nernst potential for that ion, a term that was introduced in Chapter 4 . The magnitude of the Nernst potential is determined by the ratio of the concentrations of that specific ion on the two sides of the membrane. The greater this ratio, the greater the tendency for the ion to diffuse in one direction and therefore the greater the Nernst potential required to prevent additional net diffusion. The following equation, called the Nernst equation, can be used to calculate the Nernst potential for any univalent ion at the normal body temperature of 98.6°F (37°C):

where EMF is the electromotive force and z is the electrical charge of the ion (e.g., +1 for K ⁺ ).

When using this formula, it is usually assumed that the potential in the extracellular fluid outside the membrane remains at zero potential, and the Nernst potential is the potential inside the membrane. Also, the sign of the potential is positive (+) if the ion diffusing from inside to outside is a negative ion, and it is negative (−) if the ion is positive. Thus, when the concentration of positive potassium ions on the inside is 10 times that on the outside, the log of 10 is 1, so the Nernst potential calculates to be −61 millivolts inside the membrane.

The Goldman Equation Is Used to Calculate the Diffusion Potential When the Membrane Is Permeable to Several Different Ions

When a membrane is permeable to several different ions, the diffusion potential that develops depends on three factors: (1) the polarity of the electrical charge of each ion; (2) the permeability of the membrane (P) to each ion; and (3) the concentration (C) of the respective ions on the inside (i) and outside (o) of the membrane. Thus, the following formula, called the Goldman equation or the Goldman-Hodgkin-Katz equation, gives the calculated membrane potential on the inside of the membrane when two univalent positive ions, sodium (Na ⁺ ) and potassium (K ⁺ ), and one univalent negative ion, chloride (Cl ⁻ ), are involved:

Several key points become evident from the Goldman equation. First, sodium, potassium, and chloride ions are the most important ions involved in the development of membrane potentials in nerve and muscle fibers, as well as in the neuronal cells. The concentration gradient of each of these ions across the membrane helps determine the voltage of the membrane potential.

Second, the quantitative importance of each of the ions in determining the voltage is proportional to the membrane permeability for that particular ion. If the membrane has zero permeability to sodium and chloride ions, the membrane potential becomes entirely dominated by the concentration gradient of potassium ions alone, and the resulting potential will be equal to the Nernst potential for potassium. The same holds true for each of the other two ions if the membrane should become selectively permeable for either one of them alone.

Third, a positive ion concentration gradient from inside the membrane to the outside causes electronegativity inside the membrane. The reason for this phenomenon is that excess positive ions diffuse to the outside when their concentration is higher inside than outside the membrane. This diffusion carries positive charges to the outside but leaves the nondiffusible negative anions on the inside, thus creating electronegativity on the inside. The opposite effect occurs when there is a gradient for a negative ion. That is, a chloride ion gradient from the outside to the inside causes negativity inside the cell because excess negatively charged chloride ions diffuse to the inside while leaving the nondiffusible positive ions on the outside.

Fourth, as explained later, the permeability of the sodium and potassium channels undergoes rapid changes during transmission of a nerve impulse, whereas the permeability of the chloride channels does not change greatly during this process. Therefore, rapid changes in sodium and potassium permeability are primarily responsible for signal transmission in neurons, which is the subject of most of the remainder of this chapter.

Resting Membrane Potential of Different Cell Types

In some cells, such as the cardiac pacemaker cells discussed in Chapter 10 , the membrane potential is continuously changing, and the cells are never “resting”. In many other cells, even excitable cells, there is a quiescent period in which a resting membrane potential can be measured. Table 5-1 shows the approximate resting membrane potentials of some different types of cells. The membrane potential is obviously very dynamic in excitable cells such as neurons, in which action potentials occur. However, even in nonexcitable cells, the membrane potential (voltage) also changes in response to various stimuli, which alter activities for the various ion transporters, ion channels, and membrane permeability for sodium, potassium, calcium, and chloride ions. The resting membrane potential is, therefore, only a brief transient state for many cells.

Electrochemical Driving Force

When multiple ions contribute to the membrane potential, the equilibrium potential for any of the contributing ions will differ from the membrane potential, and there will be an electrochemical driving force (V _df ) for each ion that tends to cause net movement of the ion across the membrane. This driving force is equal to the difference between the membrane potential (V _m ) and the equilibrium potential of the ion (V _eq ) Thus, V _df = V _m – V _eq .

The arithmetic sign of V _df (positive or negative) and the valence of the ion (cation or anion) can be used to predict the direction of ion flow across the membrane, into or out of the cell. For cations such as Na ⁺ and K ⁺ , a positive V _df predicts ion movement out of the cell down its electrochemical gradient, and a negative V _df predicts ion movement into the cell. For anions, such as Cl ⁻ , a positive V _df predicts ion movement into the cell, and a negative V _df predicts ion movement out of the cell. When V _m = V _eq , there is no net movement of the ion into or out of the cell. Also, the direction of ion flux through the membrane reverses as V _m becomes greater than or less than V _eq ; hence, the equilibrium potential (V _eq ) is also called the reversal potential .

Active Transport of Sodium and Potassium Ions Through the Membrane—the Sodium-Potassium (Na ⁺ -K ⁺ ) Pump

Recall from Chapter 4 that all cell membranes of the body have a powerful Na ⁺ -K ⁺ pump that continually transports sodium ions to the outside of the cell and potassium ions to the inside, as illustrated on the left side in Figure 5-4 . Note that this is an electrogenic pump because three Na ⁺ ions are pumped to the outside for each two K ⁺ ions to the inside, leaving a net deficit of positive ions on the inside and causing a negative potential inside the cell membrane.

The Na ⁺ -K ⁺ pump also causes large concentration gradients for sodium and potassium across the resting nerve membrane. These gradients are as follows:

The ratios of these two respective ions from the inside to the outside are as follows:

Leakage of Potassium Through the Nerve Cell Membrane

The right side of Figure 5-4 shows a channel protein (sometimes called a tandem pore domain, potassium channel , or potassium [K ⁺ ] “leak” channel ) in the nerve membrane through which potassium ions can leak, even in a resting cell. The basic structure of potassium channels was described in Chapter 4 ( Figure 4-4 ). These K ⁺ leak channels may also leak sodium ions slightly but are far more permeable to potassium than to sodium, normally about 100 times as permeable. As discussed later, this differential in permeability is a key factor in determining the level of the normal resting membrane potential.

Contribution of the Potassium Diffusion Potential

In Figure 5-5A , we assume that the only movement of ions through the membrane is diffusion of potassium ions, as demonstrated by the open channels between the potassium symbol (K ⁺ ) inside and outside the mem brane. Because of the high ratio of potassium ions inside to outside, 35:1, the Nernst potential corresponding to this ratio is −94 millivolts because the logarithm of 35 is 1.54, and this, multiplied by −61 millivolts, is −94 millivolts. Therefore, if potassium ions were the only factor causing the resting potential, the resting potential inside the fiber would be equal to −94 millivolts, as shown in the figure.

Contribution of Sodium Diffusion Through the Nerve Membrane

Figure 5-5B shows the addition of slight permeability of the nerve membrane to sodium ions, caused by the minute diffusion of sodium ions through the K ⁺ -Na ⁺ leak channels. The ratio of sodium ions from inside to outside the membrane is 0.1, which gives a calculated Nernst potential for the inside of the membrane of +61 millivolts. Also shown in Figure 5-5B is the Nernst potential for potassium diffusion of −94 millivolts. How do these interact with each other, and what will be the summated potential? This question can be answered by using the Goldman equation described previously. Intuitively, one can see that if the membrane is highly permeable to potassium but only slightly permeable to sodium, the diffusion of potassium contributes far more to the membrane potential than the diffusion of sodium. In the normal nerve fiber, the permeability of the membrane to potassium is about 100 times as great as its permeability to sodium. Using this value in the Goldman equation, and considering only sodium and potassium, gives a potential inside the membrane of −86 millivolts, which is near the potassium potential shown in the figure.

Contribution of the Na ⁺ -K ⁺ Pump

In Figure 5-5C , the Na ⁺ -K ⁺ pump is shown to provide an additional contribution to the resting potential. This figure shows that continuous pumping of three sodium ions to the outside occurs for each two potassium ions pumped to the inside of the membrane. The pumping of more sodium ions to the outside than the potassium ions being pumped to the inside causes a continual loss of positive charges from inside the membrane, creating an additional degree of negativity (about −4 millivolts additional) on the inside, beyond that which can be accounted for by diffusion alone.

Therefore, as shown in Figure 5-5C , the net membrane potential when all these factors are operative at the same time is about −90 millivolts. However, additional ions, such as chloride, must also be considered in calculating the membrane potential.

In summary, the diffusion potentials alone caused by potassium and sodium diffusion would give a membrane potential of about −86 millivolts, with almost all of this being determined by potassium diffusion. An additional −4 millivolts is then contributed to the membrane potential by the continuously acting electrogenic Na ⁺ -K ⁺ pump, and there is a contribution of chloride ions. As mentioned previously, the resting membrane potential varies in different cells from as low as around −10 millivolts in erythrocytes to as high as −90 millivolts in skeletal muscle cells.

Resting Stage

The resting stage is the resting membrane potential before the action potential begins. The membrane is said to be “polarized” during this stage because of the −70 millivolts negative membrane potential that is present.