426 Diffusion, Permeability, and Active Transport /23 : 3 



3. The Diffusion of Oxygen into Cells 



There are many applications of the equations developed in the last 

 section. In this chapter, three applications will be discussed. These 

 are the diffusion of oxygen in single cells, the permeability of red blood 

 cells, and the evidence for active transport across cell membranes. 



In the absence of active transport, the problem of oxygen diffusion is 

 one of finding a suitable solution to Equations 9 and 11, subject to the 

 concentration of oxygen far from the cell being held constant. In 

 mathematical terms, this is a boundary-value problem. Equation 9 is 

 studied in heat problems and in quantum mechanics. No matter what 

 significance one assigns to the symbols, Equation 9 always has the same 

 mathematical solutions. Only in certain very special geometrical 

 symmetries are mathematical solutions in a closed form 3 possible. 

 That is, in general, the problem can be solved only numerically by 

 means of lengthy computational procedures. 



From this starting point, one may take several approaches to the 

 oxygen-diffusion problem (besides the trivial one of giving up altogether). 

 The first is to restrict the discussion to cases in which permeability plays 

 the dominant role. This is pursued in Section 4. 



If effects other than permeability are considered, there still remain 

 several avenues of approach. The most exact is to set up the problem for 

 a numerical solution by a high-speed electronic computer. This has the 

 disadvantage of yielding only very specialized solutions and little or no 

 insight into the general nature of diffusion problems; it will not be 

 pursued further here. Another course is to approximate all values by 

 average ones. This is discussed briefly below. The other approach 

 considered in this section is to approximate the cell by a geometry 

 (namely spherical) in which Equation 9 can be solved. 



The Average-Value Approach 



In this approach, any arbitrary-shaped cell is approximated by either 

 a rod or a pillbox. The ends are treated as circles and the curved sides 

 are further approximated as planes. Finally, it is assumed that the 

 average concentration c exists throughout the inner part of the cell. 

 This allows one to approximate the concentration gradient at the inside 

 of the cell wall by 



8c 

 8~r 



2(*i - c) 



H 



where c x is the concentration at r x (see Figure 4). 



J That is to say, a solution exists in terms of known or tabulated functions. 



