K

FIGURE 22 Hypothetical cell containing an energy-dependent pump for KAc and leaks for K+ and Ac-.

cell will cease. But, because the pump continues to operate, in time the concentration of KAc concentration in the cell will exceed that in the extracellular solution, and then K+ and Ac— will diffuse out of the cell through their leak pathways. At a subsequent time, a point will be reached when the rate at which KAc is pumped into this cell is precisely balanced by the rates at which K+ and Ac— diffuse out of the cell; let us say, for the sake of discussion, that when this steady state is reached the concentration of KAc in the cell ([KAc]i) = 100 mmol/L. Furthermore, because the stoichiometry of the pump is one to one, the rates at which K+ and Ac— diffuse out of the cell must also be equal or the law of electroneutrality will be violated.

Because the concentration differences for K+ and Ac— across the membrane are equal—ACK = ACAc = 90 mM—if the membrane is equally permeable to K+ and Ac— (i.e., PK = PAc), equal rates of diffusion out of the cell (i.e., JK = JAc) would not pose a problem. But, what if the membrane is more permeable to the small ion, K+, than to Ac—? Then, K+ will tend to leave the cell faster than Ac— and possibly cause a violation of the law of electroneutrality. This is prevented by the establishment of an electrical potential difference across the membrane oriented such that the cell interior is electrically negative with respect to the extracellular solution. This retards the diffusion of K+ out of the cell and accelerates the outward diffusion of Ac— so that JK = JAc.

This situation is precisely analogous to that which we considered above dealing with the diffusion of KAc across an artificial membrane separating two solutions having different concentrations. The only differences are as follows:

1. In the artificial system, we provided the ''muscle'' (energy) to set up the concentration difference, whereas in our hypothetical cell, the energy is provided by the pump at the expense of ATP.

2. The artificial system will run down in time as KAc diffuses out of compartment o into compartment i, whereas in our hypothetical cell, the KAc leaving the cell is constantly being replenished by the pump; this is the essence of a steady state displaced from equilibrium by the investment of metabolic energy.

Finally, we can estimate Vm by employing Eq. 8, which describes the diffusion potential across a membrane arising from a concentration difference of a dissociable salt and differences in the permeabilities of the membrane to the resulting ions; that is,

FIGURE 22 Hypothetical cell containing an energy-dependent pump for KAc and leaks for K+ and Ac-.

(Note: In Eq. 8, we employed diffusion constants, but the use of permeability coefficients as defined in Eq. 5 is also correct.) Thus, if PK = PAc, then Vm = 0. But, if PK > PAc, then, because [KAc], > [KAc]o, Vm will be oriented such that the cell interior is electrically negative with respect to the extracellular solution. Furthermore, because ([KAc]o/[KAc]i) = 0.1 (i.e., 10 mmol/L/100 mmol/L), if PK is very much greater than PAc, then Vm will approach —60 mV, that is, the Nernst potential for K+. If PAc > PK, then the cell interior will be electrically positive with respect to the extracellular solution; in this case, Vm can approach +60 mV if PAc is very much larger than PK. Thus, Vm can assume any value between approximately +60 mV and approximately —60 mV, depending on the relation between PK and PAc. Recall that in no instance is bulk electroneutrality violated; indeed, it is Vm that assures that JK = JAc, thereby preventing a bulk separation of charge.

Now let us examine the behavior of the somewhat more realistic model illustrated in Fig. 21. Suppose that the only two ions subject to active transport are Na+ and K+ and that a one-to-one coupled carrier mechanism is involved; Cl— is assumed to cross the cell membrane only by diffusion through water-filled pores. Under steady-state conditions, (1) Na+ must diffuse into the cell at a rate equal to the rate of its carrier-mediated extrusion, and (2) K+ must diffuse out of the cell at a rate equal to the rate of its carrier-mediated uptake. If the Na+ and K+ carrier mechanism is coupled such that for every K+ pumped into the cell one Na+ is extruded, the diffusional flows of these ions in opposite directions must also be equal; otherwise, electroneutrality would be violated. Now, in the example given, the concentration differences for Na+ and K+ across the membrane are approximately equal and opposite (this is approximately the case for many cells in higher animals) so that, if the permeabilities of the membrane to Na+ and K+ were also equal, the rates of diffusion in opposite directions would be equal and electroneutrality would be preserved. However, if the membrane is much more permeable to K+ than to Na+ (as is the case for most cells), the rates of net Na+ and K+ diffusion would not be equal in the absence of a transmembrane electrical potential difference. Thus, a Vm is generated that it is oriented such as to retard the diffusion of the more permeant ion (K+) and accelerate the diffusion of the less permeant ion (Na+) so that the inward diffusion of Na+ is equal to the outward diffusion of K+. Under these conditions, the cell interior will be electrically negative with respect to the exterior. To repeat, a diffusion potential arises across the membrane that maintains equal diffusion rates and preserves electroneutrality despite the fact that the permeabilities of the membrane to the two ions are not equal.

Several important points should be noted:

1. The example dealing with Na+ and K+ diffusion in opposite directions is formally analogous to our earlier example (Fig. 22) in which K+ and Ac— diffused in the same direction, because, as far as electroneutrality is concerned, the flow of a cation in one direction is equivalent to the flow of an anion in the opposite direction. In both examples, when K+ is the more permeable ion, the orientation of Vm will be the same (i.e., so as to retard the flow of K+ and accelerate the flow of the less permeable ion).

2. The electrical potential differences (Vm) described in both examples are diffusion potentials resulting from the diffusional flows of ions down concentration differences. Although Vm is dependent on ion pumps, the relation is indirect; the pumps merely serve to establish the ionic concentration differences that provide the driving forces for the diffusional flows.

The orientation of Vm is always such as to retard the diffusional flow of the more permeant ion and accelerate the flow of the less permeant ion, and the magnitude of Vm is dependent on the individual permeabilities and concentrations of these ions (because these are the direct determinants of the diffusional flows that must be equalized by this electrical potential difference). We can generate an expression that defines the magnitude and orientation of Vm using an argument similar to that employed when we considered the diffusion potential generated by KAc diffusion out of the hypothetical cell shown in Fig. 22. Thus, if PK is much greater than PNa, then Vm will approach the Nernst potential for K+—that is, 60 log ([K +]o/[K+]i)—and, because [K+]o < [K+],-, the cell interior will be electrically negative with respect to the extracellular compartment (i.e., Vm < 0). Conversely, if PK is much less than PNa, then Vm will approach the Nernst potential for Na+—that is, 60 log ([Na+]o/[Na+],)—and, because [Na+]o > [Na+],-, under these conditions Vm > 0. These two extremes are satisfied by Eq. 26, which is sometimes referred to as a double Nernst equation:

This equation can be rewritten in the form:

where a = (PNa/PK). When PK is much greater than PNa, a is very small and Vm approaches the Nernst potential for K+; conversely, when PK is much less than PNa, a is very large and Vm approaches the Nernst potential for Na+. Thus, Vm can have any value between these two extremes depending on the value of a.

An expression having the form of Eq. 26 was first formally derived by Goldman in 1943. It was reder-ived by Hodgkin and Katz in 1949 and, discussed in Chapter 4, was first applied successfully to the analysis of the ionic basis of the resting and action potential of the squid axon. During the past three decades, it has provided the basis for understanding the origin of Vm across a wide variety of biologic membranes.

Finally, it should be noted that the stoichiometry of the (Na-K) pump is three Na+ for two K+ (not one to one). All this means is that, when a steady state is achieved, three Na+ ions must diffuse into the cell for every two K+ ions that diffuse out of the cell. But, if PNa < PK the final result will still be that Vm is oriented such that the cell interior is electrically negative with respect to the extracellular fluid.

Before concluding this section, let us consider the distribution of Cl" resulting from this pump-leak system. If, as stipulated above, the Cl" distribution is determined solely by diffusion, then, when there is no net movement of Cl" across the membrane (i.e., the composition of the cell is constant), the chemical potential forces acting on Cl" must be balanced by the electrical forces acting on this anion. Thus, if Vm is negative (with respect to the outer solution), then the intracellular concentration of Cl" (i.e., [Cl]() will be less than that in the extracellular solution (i.e., [Cl]o). The precise relation among Vm, [Cl]i, and [Cl]o is given by the Nernst equation, which, as discussed above, describes the condition for the balance or equality of chemical and electrical forces:

When Vm is negative, the right-hand side of Eq. 29 has a value less than 1 so that [Cl]; < [Cl]o. Thus, if Vm is approximately —60 mV, then [C1], will be approximately one-tenth [Cl]o.

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