23 : 3/ Diffusion, Permeability, and Active Transport 



427 



It is further necessary to assume that in a short distance 8 the con- 

 centration c' outside the cell reaches the value c' it has at long distances. 

 Then the concentration gradient outside 

 the cell wall may be approximated as 



dr 



c - c x 



where r' is the average cell radius. 



With some juggling, it is possible to 

 rewrite Equations 9 and 11 as 



+ q 



k( Cl - c[) = D 



,Cl 



2D C<1 ' 



D' C -^- 



= k(c 2 - 4) 



where D' is the diffusion constant outside 

 the cell. 



These terms may be further approximated 

 to show that for the case 



q = constant 

 one may write 



c = c + Xq + A\e~ tu 

 f- 1 -& c o + *<i + ^ c-—^^ + Xq (12) 



(b) 



Figure 4. The average value 

 approach. An arbitrary cell 

 is shown in Figure (a) with 

 average radii r x and r 2 . It is 

 assumed that the concentration 

 in the central part of the cell is 

 constant reaching the value of 

 the average concentration c at 

 r 1 /2 and r 2 /2 from the center. 

 It is further assumed that the 

 cell can be replaced by a rod 

 or pillbox as shown in Figures 

 (b) and (c). The general 

 patterns of diffusion which 

 do not depend on exact cell 

 geometry are then investigated. 

 After N. Rashevsky, Mathe- 

 matical Biophysics (Chicago : 

 University of Chicago Puss, 

 1948). 



In this expression, c and A are compli- 

 cated constants involving the parameters 

 k, D, /_)', r 1} and r 2 . This entire approach 

 is developed in detail in the reference by 

 Rashevsky; it is hoped that more mathe- 

 matically inclined readers will refer to this text. 



The approximate expression, Equation 12, emphasizes that after a 

 sufficient length of time a steady state will be set up if q is time indepen- 

 dent. The preceding solution also shows the form of result to expect 

 from any approximation. Comparing Equation 1 2 with exact expressions 

 for spheres allows one to demonstrate that the exact cell shape is not 

 important. 



