Addendum A

Equilibrium Potentials

The following example helps illustrate the concept of an equilibrium potential, as discussed in the lecture on 11/14. You do not need to memorize this example. Still you may find it a useful way to visualize the underlying concepts.

Pictured above are two adjoining aqueous compartments. They are separated by an artificial membrane that is permeable to K⁺ ions, and impermeable to Cl^- ions and H₂O. At the beginning of the experiment (t = 0), Compartment #1 is filled with a 100 mM solution of KCl and Compartment #2 is filled with a 50 mM solution of KCl. [Note: in water, KCl dissociates completely into K⁺ ions and Cl^- ions.]

1. The flow of K⁺ ions through the central membrane will be determined by the summed vectors of [i] diffusion down the K⁺ concentration gradient and [ii] any electrical force acting on the K⁺ ions as they cross the membrane.

2. At t = 0, there is no electrical potential difference between the two compart-ments and no electrical force at the membrane. Thus, the initial rate of K⁺ flow is determined solely by the concentration difference, with a net flow of K⁺ ions passing from Compartment #1 to Compartment #2.

3. As time proceeds, the uncompensated flow of K⁺ ions will make Compartment #2 more positive and Compartment #1 more negative, producing an electrical gradient across the central membrane. This electrical gradient will begin to counteract the concentration gradient and retard the flow of additional K⁺ ions into Compartment #2.

4. One can visualize this phenomenon by graphing the rate of K⁺ ion flow and the increasing electrical potential between the two compartments as a function of time.

As shown in these graphs, the rate of K⁺ ion flow will drop to 0 and the membrane potential will rise to a plateau value with the exact same exponential time-course. Once they reach their plateau values, the system is at equilibrium.

5. The electrical potential across the membrane at this equlibrium is called a K⁺ equilibrium potential.

6. Quantitative calculation of an equlibrium potential can be greatly simplified by taking advantage of the following generality: in most biological system, the number of ions that must cross a membrane to produce the equilibrium electrical potential is negligible compared to the concentration of ions in either compartment. Thus, one can ignore any changes in ion concentration and use the starting ion concentrations to calculate the equilibrium potential that will be reached. For the example shown here, the K⁺ equlibrium potential can be calculated by the so-called Nernst equation:

E_K = 58 log₁₀ ([K⁺]₁/[K⁺]₂)

in which:

E_K is the K⁺ equlibrium potential in millivolts (Chamber #2 minus Chamber #1).

[K⁺]₁ is the concentration of K⁺ ions in Chamber #1.

[K⁺]₂ is the concentration of K⁺ ions in Chamber #2.

For the example discussed here:

E_K = 58 log₁₀ (100 mM/50 mM) = 58 log₁₀ (2) = 17.4 mV

7. Note that the K⁺ equilibrium potential represents the electrical gradient necessary to counterbalance a particular K⁺ concentration gradient. Hence, the actual value of the equlibrium potential depends on the specific K⁺ concentrations in the two compartments.

Also note that there is nothing special about K⁺ ions. This same line of reasoning can be applied to any ion species that can pass between compartments through a semi-permeable membrane.