.23 : 4/ Diffusion, Permeability, and Active Transport 431 



c M may be set equal to zero. If, in addition, there are no penetrating 

 solutes, S and c s will also be zero. Equation 15 can also be simplified 

 for solutes penetrating far more rapidly than water. Such solutes can 

 be described by c s = SjV because a rapidly penetrating solute will 

 equilibrate before a change in water occurs. In either case, that is, 

 no penetrating solutes or very rapidly penetrating ones, Equation 18 

 may be approximated by 



dV (c V \ 



dt ~ ^ W \~V) 



which can be integrated directly to give 



V 2 = V 2 + 2P w c V t (19) 



Equation 19 may be used to find the value of P w for erythrocytes. The 

 value per unit surface area is 10 to 30 times greater than for most other 

 biological cells. This high value will be commented on further in the 

 next section. 



Because the value of P w is so high for red blood cells, it follows that 

 most solutes will penetrate more slowly than water. Then, one may 

 solve Equation 1 7 for the penetration of the solute, assuming that the 

 water equilibrium is attained instantaneously. This is equivalent to 

 the assumption that the external concentration c s is always iso-osmotic 

 with c , the original concentration within the cell (provided that c M is 

 zero) Accordingly, one may write 



S= Cs (V - V ) or dS=c s dV 



This last differential may be substituted into Equation 17, describing 

 the flow of s into the cell. The variable S disappears, giving 



dj_ _ p c_sVo 

 dt' V 



On integrating, this becomes 



V 2 = V 2 - Pc s V t (20) 



Using this relationship, the permeability P may be found for all solutes 

 which penetrate less rapidly than water. 



Equations 19 and 20 allow one to compute the time for 90 per cent 

 saturation of the cell, assuming that diffusion within the cell is sufficiently 

 rapid that the concentration within the cell is constant. Similarly, 

 Equation 5 could be used, assuming that P was infinite. Such calcula- 

 tions have been carried out for a number of different solutes. It was 

 found that the times for intracellular diffusion are negligible compared 

 to the time for permeating the cell membrane. 



