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The calculation for the amount of K+ that had to diffuse across the membrane to establish the Vm of -80 mV is as follows: 1 mole of K+ contains 96,480 coulombs, so 1.05 × 10^-17 moles of K+ needed to diffuse.

The total intracellular K+ content, with a cell volume of 4.19 × 10^-12 L and an intracellular [K+] of 120 mmol/L, is 5.03 × 10^-13 moles. Therefore, the diffusion of 1.05 × 10^-17 moles of K+ out of the cell represents only a 0.002% change in the intracellular K+ content.

Thus, the intracellular [K+] of the cell is not appreciably altered by the diffusion of K+ out of the cell.

Cells have developed mechanisms to regulate their volume. Most cells are highly permeable to water due to aquaporins in their plasma membranes. Osmotic pressure gradients across the cell membrane, generated by effective osmoles, cause water to move into or out of the cell, resulting in changes in cell volume. Cells swell in hypotonic solutions and shrink in hypertonic solutions. Even in an isotonic solution, the maintenance of cell volume requires ATP expenditure and the activity of the Na+,K+-ATPase.

The Na+,K+-ATPase is crucial for isotonic cell volume regulation. Red blood cells swell when chilled or when the Na+,K+-ATPase is inhibited by cardiac glycosides such as ouabain and digoxin (Lanoxin). Energy expenditure is necessary to maintain cell volume in an isotonic solution due to the effect of intracellular proteins on ion distribution across the plasma membrane, known as the Gibbs-Donnan effect.

The Gibbs-Donnan effect occurs when a membrane separating two solutions can be permeated by some but not all molecules in solution. This effect accounts for the small differences in ionic compositions between plasma and interstitial fluid. In cells, the plasma membrane acts as the membrane, and the impermeant molecules are the intracellular proteins and organic molecules.

The presence of impermeant molecules in one compartment, such as proteins, leads to the accumulation of permeant molecules/ions in the same compartment over time. This increases the number of osmotically active particles and the osmotic pressure, causing water to enter the compartment. In cells, the Gibbs-Donnan effect would increase the number of osmotically active particles inside the cell, resulting in cell swelling. However, the activity of the Na+,K+-ATPase counteracts this effect by actively extruding cations (three Na+ ions are extruded while two K+ ions are brought into the cell).

Furthermore, the K+ gradient established by the Na+,K+-ATPase allows for the development of Vm, where the cell's interior is electrically negative. This drives Cl- and other anions out of the cell. Through the activity of the Na+,K+-ATPase, the number of intracellular osmotically active particles is regulated.

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Because1moleofK+ contains96,480coulombs,theamountofK+ thathadtodiffuseacrossthemembranetoestablishtheVmof−80mViscalculatedasfollows: . coul = 1 05. × 10 mole ofK , ombs/mol
Withacellvolumeof4.19× 10−12L(volume= 4πr3/3)andanintracellular[K+]of120mmol/L,thetotalintracellularK+ contentis 4 19 × 10 × . mol/L = 5 03 . 012 . × 10 moles
Therefore,thediffusionof1.05× 10−17molesofK+ outofthecellrepresentsonlya0.002%changeintheintracellularK+ content: . mol −13 ° 0 002. % 5 03 × 10 es . mol
Thustheintracellular[K+]ofthecellisnotappreciablyalteredbythediffusionofK+ outofthecell.
Regulation of Cell Volume
As already noted, changes in cell volume can lead to cell damage and death. Cells have developed mechanisms to regulate their volume. Most cells are highly permeable by water because of the presence of aquaporins in their plasma membranes. As discussed in
Chapter 1, osmotic pressure gradients across the cell membrane that are generated by effective osmoles cause water to move either into or out of the cell, which result in changes in cell volume. Thus cells swell when placed in hypotonic solutions and shrink when placed in hypertonic solutions (see the section ”). However, even when a cell is placed in an isotonic solution, the maintenance of cell volume is an active process requiring the expenditure of ATP and specifically the activity of the Na+,K+-ATPase.
The importance of the Na+,K+-ATPase in isotonic cell volume regulation can be appreciated by the observation that red blood cells swell when chilled (i.e., reduced ATP synthesis) or when the Na+,K+-ATPase is inhibited by cardiac glycosides (e.g., ouabain, digoxin [Lanoxin]). The necessity for energy expenditure to maintain cell volume in an isotonic solution is the result of the effect of intracellular proteins on the distribution of ions across the plasma membrane: the so-called Gibbs-Donnan effect (
Fig. 2.6
The Gibbs-Donnan effect occurs when a membrane separating two solutions can be permeated by some but not all of the molecules in solution. As noted previously, this effect accounts for the small differences in the ionic compositions of the plasma and the interstitial fluid. In this case, the capillary endothelium represents the membrane, and the plasma proteins are the molecules whose ability to permeate across the capillary is restricted. For cells, the membrane is the plasma membrane, and the impermeant molecules are the intracellular proteins and organic molecules.
Fig. 2.6 , the presence of impermeant molecules (e.g., protein) in one compartment results over time in the accumulation of permeant molecules/ions in the same compartment. This increases the number of osmotically active particles in the compartment containing the impermeant anions, which in turn increases the osmotic pressure, and water thereby enters that compartment. For cells, the Gibbs-Donnan effect would increase the number of osmotically active particles in the cell, and result in cell swelling. However, the activity of the Na+,K+-ATPase counteracts the Gibbs-Donnan effect by actively extruding cations (three Na+ ions are extruded, whereas two K+ ions are brought into the cell). In addition, the K+ gradient established by the Na+,K+-ATPase allows for the development of the Vm (in which the cell’s interior is electrically negative), that in turn drives Cl− and other anions out of the cell. Thus through the activity of the Na+,K+-ATPase, the number of intracellular osmotically active particles is