Osmotic Volume Flow in a Donnan System with Permeant Ions

Impermeant molecules inside a cell would lead to an inward osmotic flow of water, causing swelling, were it not for the pumping of permeant sodium ions out of the cell as soon as they leak in. The energy barrier model for a semipermeable membrane, first introduced by Debye to provide a molecular-level explanation of the van’t Hoff equation for osmotic pressure, can be used to advantage in this situation, since the pump can be conceptualized as increasing the energy barrier for the sodium ion. The Debye model has previously been extended to include osmosis induced by electrostatically neutral solutes. Discussion of the effect of ion pumping on water transport requires an understanding of osmosis in systems containing permeant ions, that is, Donnan systems. We have obtained an equation for Donnan osmosis across a Debye energy barrier that separates an aqueous solution of permeant sodium, potassium, and chloride ions from a solution containing these permeant ions and additionally an impermeant anion, the latter representing intra-cellular impermeant charged species. Donnan osmosis occurs even if osmolarities on the two sides of the membrane are equal. Numerical representation shows that the Donnan-Debye model provides a quantitative theoretical framework for the action of the sodium/potassium/ATPase ion pump as effectively rendering the extracellular sodium ions impermeant, thus balancing the impermeant molecules inside the cell. Another application of Donnan osmosis shows that ion charge effects, missing from lists of Starling forces, are nonetheless expected to be a major contributor to transport across capillary walls.