A THEORY OF THE SURVIVAL VALUE OF MOTILITY 139 



A MODEL SYSTEM 



To understand the energetics of motility it is convenient to con- 

 sider a model system which possesses the essential characteristics of a 

 single microorganism and at the same time lends itself to a formal 

 analysis. We take as our model a single spherical metabolizing cell, 

 in a liquid bath containing all the substances essential for life and 

 growth. The cell is assumed not to require mating for growth, and 

 the environment is taken as free of predators and noxious agents. We 

 then ask, how does the rate of food (or energy) collection and the 

 rate of energy expenditure depend on the average velocity with which 

 such a cell moves through the liquid when the motion is such that 

 the Reynolds number of the system is much less than one? 



THE STATIONARY METABOLIZING SPHERICAL CELL 

 IN AN UNSTIRRED BATH 



Consideration will be given first to the case of the stationary me- 

 tabolizing spherical cell in an unstirred bath in order to gain some 

 understanding of the factors that determine the rate of food collec- 

 tion in the simplest possible case. The only process by which nutrient 

 substances can get to such a cell is by diffusion, of course. The con- 

 centration of those substances metabolized by the cell is reduced at 

 the surface of the cell, and a concentration gradient is established. 

 The gradient develops both in space and time until a steady state of 

 diffusion is reached in which nutrient molecules diffuse to the cell 

 surface at a rate just equal to that at which they are metabolized by 

 the cell. Since the lowest value that the concentration of any sub- 

 stance can have at the surface of the cell is zero, the maximum 

 steady state rate of diffusion to the cell surface will be obtained for 

 this condition. Our problem then is to develop from diffusion theory 

 the relation for the steady state rate of flux of diffusing substance 

 across a spherical surface of given radius. This is the problem of the 

 spherical sink that is treated in almost every text on diffusion processes. 

 To solve it one considers a stationary sphere of radius R at the origin. 

 At time zero, the concentration of the diffusing substance is taken as 

 zero for all values of the radial coordinate r, less than or equal to R, 



