The Na/K pump, Cl ion, and osmotic stabilization of cells

Abstract

An equation for membrane voltage is derived that takes into account the electrogenicity of the Na/K pump and is valid dynamically, as well as in the steady state. This equation is incorporated into a model for the osmotic stabilization of cells. The results emphasize the role of the pump and membrane voltage in lowering internal Cl(-) concentration, thus making osmotic room for vital substances that must be sequestered in the cell.

Figure 1

The model cell in normal conditions. Its membrane contains three types of channels, Cl, K, and Na, which mediate passive movement of these ions. Ion concentrations are in mM. Permeability of the resting cell is P_K 1.0, P_Na 0.02, and P_Cl 2.0 (in arbitrary units). An ATP-driven pump extrudes three Na⁺ ions and imports two K⁺ ions per cycle. The resting potential is −80 mV, and the resting volume is 1.0 (arbitrary units). Internal [Cl⁻] is only 7 mM, making osmotic room for internally sequestered substances, S⁻, at a concentration of 143 mM.

Figure 2

Theoretical current–voltage relations for open Na and K channels. (A) The solid curves are from Eqs. 3a (the Na curve is the almost horizontal line, just negative to zero), and the dashed curves from Eqs. 3b. For both sets of curves, I_Na = −I_K at −80 mV; i.e., the resting potential is −80 mV. (B) Approximately linear current–voltage relations, generated from Eqs. 8. These relations are most similar to the experimentally observed curves, but the model works well when any of the three pairs of current–voltage relations are used, as described in the text.

Figure 3

Cl permeability and membrane voltage. Traces were generated by the model in the text to simulate a classical experiment of Hodgkin and Horowicz (13). (A) To the left of 0 on the time axis, the cell is in normal conditions. At 0 min, [K⁺]_o was increased from 5 to 20 mM, causing a quick change of V_m. A slower change follows as Cl⁻, no longer at equilibrium, drifts into the cell, causing a small volume increase. During the interval when Cl⁻ is not in equilibrium (0 min to ≈9 min), there is a Cl⁻ current that keeps V_m negative to its steady-state level, −50 mV. The current weakens with time as equilibrium ([Cl⁻]_i = 23 mM, V_m = −50 mV) is approached. (B) The experiment is repeated after setting P_Cl to zero at the time indicated. When [K⁺]_o is increased to 20 mM at time 0, V_m rises within milliseconds to −50 mV. The time course is quicker than in A because there is no Cl⁻ current. Notice that [Cl⁻]_i, the very bottom trace, and volume do not change.

Figure 4

The pump, Cl⁻, and cell volume. (A) Turning off the pump of an unperturbed cell leads to an exchange of Na⁺ for K⁺ internally. As V_m rises, Cl⁻ is no longer in equilibrium and enters the cell. The volume increases as the result of increased cell content of Na⁺ and Cl⁻ (Na⁺ enters faster than K⁺ is lost). Swelling continues, in theory, until volume is infinite. (B) P_Cl is decreased to zero at the arrow. When the pump is turned off, the changes are similar to those in A except that Cl⁻ cannot enter and volume remains constant. On restoring P_Cl to 2, V_m jumps negative, pulled by Cl⁻ current, then moves toward zero as Cl⁻ equilibrates. The negative jump of V_m causes a transient increase in [K⁺]_i as internal negativity pulls K⁺ inward. The quick volume increase causes a transient dilution of Na⁺. Volume then increases slowly toward infinity, as Na⁺ and Cl⁻ enter the cell. The total cell content of S⁻ remains fixed, but [S⁻] falls as volume increases.