Inasmuch as attempts to explain accumulation by the Donnan principle have failed in the case of Valonia, a hypothesis of the steady state has been formulated to explain what occurs. In order to see whether this hypothesis is in harmony with physico-chemical laws attempts have been made to imitate its chief features by means of a model.
The model consists of a non-aqueous layer (representing the protoplasmic surface) placed between an alkaline aqueous phase (representing the external solution) and a more acid aqueous phase (representing the cell sap).
The model reproduces most of the features of the hypothesis. Attention may be called to the following points.
1. The semipermeable surface is a continuous non-aqueous phase.
2. Potassium penetrates by combining with an acid HX in the non-aqueous layer to form KX which in turn reacts with an acid HA in the sap to form KA. Since KX is little dissociated in the non-aqueous layer potassium appears to pass through it chiefly in molecular form.
3. The internal composition depends on permeability, e.g., sodium penetrates less rapidly than potassium and in consequence potassium predominates over sodium in the "artificial sap." The order of penetration in the model is the same as in Valonia, i.e., K > Na > Ca > Mg, and Cl > SO4, but the quantitative resemblance is not close, e.g., the difference between potassium and sodium, and chloride and sulfate is much less in the model.
4. The formation of KA and NaA in the sap raises its osmotic pressure and water enters.
5. The concentration of potassium and sodium and the osmotic pressure become much greater inside than outside. For example, potassium may become 200 times as concentrated inside as outside.
6. No equilibrium occurs but a steady state is reached in which water and salt enter at the same rate so that the composition of the sap remains constant as its volume increases.
7. Since no equilibrium occurs there is a difference of thermodynamic potential between inside and outside. At the start the thermodynamic potential of KOH is much greater outside than inside. This difference gradually diminishes and in the steady state has about the same value as in Valonia. The difference in pH value between the internal and external solutions is also similar in both cases (about 2 pH units).
8. Accumulation does not depend on the presence of molecules or ions inside which are unable to pass out.
One important feature of the hypothesis is not seen in the model: this is the exchange of HCO3 for Cl-. Experiments on this point are in progress.
The jamming transition in high dimension: an analytical study of the TAP equations and the effective thermodynamic potential