126 L. J. MULLINS 



There are still, however, only a limited number of such arrangements because 

 of the small number of water molecules comprising the second layer of hydra- 

 tion. If K + approaches a pore that is precisely the same size as this ion with its 

 first hydration shell (denoted (K + )i), it may, as indicated previously, exchange 

 hydration, for water shells from 2 to infinity, for a similar attraction with the 

 structure lining the pore. If the pore is somewhat smaller than (K + )i , penetra- 

 tion cannot occur for steric reasons, while if the pore is somewhat too large, 

 penetration likewise cannot occur because the attraction of the ion for water 

 shells of 2 and greater is not compensated by a solvation of similar magnitude 

 in the pore. Such solvation demands a close fit between the ion and the pore 

 wall. 



Since a collection of pores formed by cylindrical macromolecules undergoing 

 thermal motion may be expected to show a dispersion in size, we have in this 

 model a mechanism for discriminating between Na + and K + that resides solely 

 in the size of the ion. Sodium ion is usually considered to be larger than K + , 

 because its larger energy of hydration results in a lower mobility in aqueous 

 solutions; in the present discussion it suffices to note that while the difference 

 in hydration energy for shells greater than (Na + ) 2 or (K + ) 2 is 1 kcal/mole, and 

 the difference between (Na + )i and (K + )i is 5 kcal/mole, a similar enhancement 

 in pore solvation energy may be expected for Na + . Because the crystal radius 

 of Ca ++ is the same as that for Na + , these ions will have very close to the same 

 size at all levels of hydration, hence Ca ++ can be considered as a competitor for 

 Na+ pores. Values for other ions are: (Mg++)i 3.37, (Mg++) 2 6.0, (Cl~)i 4.56 A. 

 radius. 



In order to account for the resting potential of axons under a variety of con- 

 ditions, Hodgkin and Katz (1949) have proposed that the permeabilities of 

 ions concerned with the potential stand in the following ratios: P K = 

 1.0, P Na = 0.04, and P c i = 0.45. Since the permeability of the membrane to an 

 ion is a function of the product, partition coefficient X mobility, we might, for 

 the moment, make the simplifying assumption that all three ions are of equal 

 mobility in the membrane, and that the number of pores of an appropriate size 

 to fit a particular ion is equivalent to the partition coefficient of this ion. In 

 Fig. 2 we have plotted the normal distribution curve that is specified by the 

 permeabilities and radii of (Na + )i , (K + )i , and (Cl~)i . The assumption has 

 been made that a 'good fit' for ion solvation purposes is 0.05 A.±, although 

 some permeability data on the red cell where Rb+ competes with K + suggest 

 that ±0.1 A might be a better value. The question of the extent to which the 

 membrane potential is dependent upon a particular value of mean interspace 

 size, can be answered by shifting the distribution curve to larger and smaller 

 mean values and computing the partition coefficient for each of the ions as well 

 as the expected membrane potential, using the figures for squid axon (Hodgkin 

 and Katz, 1949). The results of such calculations are shown in Fig. 3 and indi- 



