23 : 2/ Diffusion, Permeability, and Active Transport 



425 



by a permeability k such that the mass per unit time passing through the 

 membrane is given by 



1 dm 



A 8t 



"57 = k ^ 



'3 / | membrane 



(10) 



This in turn must equal the mass flux entering and leaving the membrane. 

 The equations describing this are 



J>i 



8c 1 



8x 



^3— 3 



3 dx 



(11) 



x = 



x-h 



Equations 9 and 1 1 , with the 

 proper values for the diffusion 

 coefficient D, the permeability 

 k, and the rate of generation of 

 the substance q, completely 

 define all biological diffusion 

 problems, from a mathematical 

 point of view. 2 Although in 

 principle they can always be 

 solved, in practice this is not 

 always easy. In the remainder 

 of the chapter, certain examples 

 are worked out. However, there 

 are two restrictions to Equations 

 9 and 10 which should be noted. 



First, it has been assumed that stirring did not occur. If random or 

 turbulent stirring is present, one may include it by using appropriately 

 larger values for D. It also has been implicitly assumed in the foregoing 

 derivation that no electrical potential gradients are present. Most 

 protoplasm conducts electrical charge so well that potential gradients 

 can exist only across the membranes. If these are present, Equation 1 1 

 must be modified, replacing the middle expression by 



Figure 3. Idealized membrane. This is used 

 in permeability discussions. The membranes 

 around many cells and subcellular structures 

 are actually three layers thick. The outer 

 two layers are believed to be protein mono- 

 layers, whereas the central layer is phospho- 

 lipid about two molecules thick. 



-k\c 1 



,-zFVIRT 



where z is the charge on the ion, F the Faraday, R the gas constant per 

 mole, T the absolute temperature, and Fthe electrical potential difference 

 across the membrane. This extra factor occurs because the partial molal 

 free energies inside and outside differ by zFV when the concentrations 

 are equal. 



2 Throughout this chapter, the purist will insist that it would be better to use 

 activities than concentrations. Although this is true, there appears to be little 

 advantage in this distinction in the examples discussed in this chapter. 



