## Transmembrane Distribution of Solutes Under Steady State Conditions

The above considerations provide us with the principles that permit us to deduce important information with regard to the nature of the distributions of metabolically inert solutes across cell membranes when the cell composition is constant (i.e., in a steady state). Thus, let us assume that we can measure the electrical potential difference across a cell membrane, Vm, and at the same time determine the intracellular and extracellular concentrations (or, more properly, activities) of any solute i (i.e., C\ and Co, respectively).

Now, the Nernst equation provides us with the criterion for determining whether the ratio of the concentrations (activities) of i across the membrane can be attributed entirely to thermal (passive) forces or whether additional (active) forces are necessary. Thus, if (at 37°C),

Vm = Ei = (60/zj) log[Co/Ci](mV) (30) or, multiplying both sides of Eq. 30 by zh zVm = 60log[Co/Ci ](mV) (31)

then, the steady-state distribution of i across the membrane can be considered passive. If this equality does not hold, then forces in addition to thermal energy must be involved in establishing the observed distribution ratio of i.

Now, we can rearrange Eq. 31 to provide the somewhat more useful expressions:

Thus, if we know Vm and the extracellular concentration of solute i, Cio, we can predict the intracellular concentration Cii that would be consistent with a passive distribution of i across the membrane and then compare that predicted value with the actual measured value.

For a neutral solute (i.e., zt = 0), Eq. 32 states that the distribution of i across the cell membrane is independent of Vm and that if Cii = Cio, this distribution can be accounted for by passive transport processes that do not require direct or indirect coupling to a source of metabolic energy. If Ci > Co, then metabolic energy must be invested into the transport process to pump i into the cell. Conversely, if C\ < Co, then energy must be invested to extrude i from the cell. Thus, for a neutral, inert solute, Cii = Cio, transport across the cell membrane cannot be attributed to diffusion or carrier-mediated, facilitated diffusion.

Now let us turn to charged solutes and illustrate this approach by considering a cell bathed by a plasma-like solution, where [Na+]o = 140 mmol/L, [K+]o = 4 mmol/L, [Cl-]o = 120 mmol/L, and [Ca2+]o = 2 mmol/L. The electrical potential difference across this membrane is determined to be -60 mV, and the intracellular concentrations of Na+, K+, Cl-, and Ca2+ are determined to be 10, 120, 12, and 10-3 mmol/L, respectively.

With respect to Na+, Eq. 32 predicts that the intracellular concentration consistent with a passive distribution should be [Na+],- = 10[Na+]o = 1400 mmol/L, but the observed value of [Na+],- was only 10 mmol/L. Thus, the distribution of Na+ across the cell membrane cannot be attributed to passive transport processes. Instead, energy must be invested by the cell to extrude Na+ and thereby lower its intracellular concentration to a level well below that predicted for a simple passive distribution.

Turning to K+, Eq. 32 predicts that if K+ is passively distributed across the membrane, its intracellular concentration should be [K^ = 10[K+]o = 40 mmol/L. The observed intracellular concentration (120 mmol/L) is much greater than this predicted value so that energy must be invested to actively pump K+ into the cell.

With respect to Cl-, the predicted value for [Cl-],- is 0.1 [Cl-]o, or 12 mmol/L. This agrees with the actual measured value so that one can conclude that the distribution of Cl- across the cell membrane is the result of passive transport processes that do not require an investment of energy on the part of the cell.

Finally, applying Eq. 32 to the case of Ca2+, we see that its predicted intracellular concentration is [Ca2+]i = 2 x 102 [Ca2+]o = 200 mmol/L. This predicted value is much greater than the observed value of only 10-3 mmol/L so that the cell must invest energy into active transport processes to extrude Ca2+ from its interior. In short, the application of the Nernst equation, which defines the thermal balance of chemical and electrical forces across a membrane, permits us to determine whether the distribution of any inert solute across a cell membrane is passive or active.

Two caveats with regard to the application of the Nernst equation to the steady-state distributions of solutes across cell membranes must be noted. First, nonconformity with the predictions of the Nernst equation simply means that the observed distribution of a solute cannot be attributed to passive forces alone, but such nonconformity provides no insight into the detailed mechanism(s) responsible for this active distribution. Second, this line of reasoning does not apply to solutes that are either produced or utilized by cells. For example, the steady-state glucose concentration in most cells is much lower than that in the extracellular fluid because this nutrient is rapidly metabolized after gaining entry into the cells. As another example, the steady-state concentration of urea in most cells is greater than that in the extracellular fluid inasmuch as it is produced by protein catabolism and exits the cell by passive transport processes; thus, under steady-state conditions, there must be a concentration difference across the membrane that provides the driving force for urea exit at a rate equal to that at which it is produced. Clearly, an uncritical application of the criteria discussed above to these two neutral solutes would suggest that glucose is actively extruded from the cell and that urea is actively accumulated by these cells.

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