424 Diffusion, Permeability, and Active Transport /23 : 2 



In vector notation, Equation 5 becomes 



| = DV, (8) 



Equations 8 and 5 as well as Equation 3 are often called Pick's diffusion 

 equation or Ficks law. Equation 8 is also referred to by physicists and 

 chemists as "the heat equation" because its form is identical to that for 

 the variation of temperature as a function of space and time in the 

 absence of any sources or sinks 1 of heat. Although the substance con- 

 sidered in Equations 3 and 5 was called oxygen for convenience, there 

 is nothing that restricts the use of these equations to oxygen. They 

 are restricted, however, to the case in which the substance is not being 

 either generated or used up chemically. 



Actually, oxygen is almost always either being used or being liberated 

 within a living cell. In most cells it is being used, and in others that 

 are photosynthesizing, it is being produced. If the rate of production 

 per unit volume is called q, Equation 8 must be replaced by 



| = DVh + q (9) 



A negative value of q implies use of oxygen (or, at any rate, of the sub- 

 stance to which the equation applies). In general, the production rate 

 q may vary with x, y, z, and t in any arbitrary manner. Equation 9 is 

 known as the inhomogeneous diffusion equation. 



Equation 9 is the final form of the diffusion equation which is developed 

 here. It can be used to analyze many different types of situations of 

 biological significance, some of which will be considered in subsequent 

 sections of this chapter. However, Equation 9 is not in a form which is 

 useful at cell membranes. The remainder of the current section is 

 devoted to deriving suitable expressions, describing mathematically the 

 permeability of cell membranes. These, when combined with Equation 

 3, 5, or 9 as appropriate, are used in the problems considered in the 

 remainder of this chapter. 



Across the membrane there may be a very large change in concentra- 

 tion. In theory, the membrane and surrounding fluids could be treated 

 as three regions, as indicated in Figure 3. In each region there will be 

 a different diffusion constant. By and large, the membranes are so 

 thin that no empirical meaning can be assigned to the concentration c 2 

 within the membrane. Instead, the membrane is usually characterized 



1 A heat sink is a place where heat energy is removed (of course being converted 

 into some other type of energy). 



