140 FRANCIS D. CARLSON 



and it is taken as C for all values of r greater than R. The solution 

 to the diffusion equation for this case is 



C = C 



1 - - + 



/ exp ( — .v 2 ) dx 



R 9R (r-R)l2(D n t) 



>"Vtt 



where C is the concentration, D n the diffusion coefficient of the nu- 

 trient, and t the time. 



The flux <i> across the surface r = R is 



<J> = 4tZ)„ (V 2 d A = 4rDJRC* (l + ^^72) 



as 1 _> 00, $ — > * s = 4tt Z) ft • R • Co , the steady-state flux of nutrient 

 across the cell boundary. 



The significance of this result should be examined before proceed- 

 ing further. The rate at which the stationary cell collects a particu- 

 lar nutrient from an unstirred medium depends directly on the con- 

 centration of the nutrient, C , the diffusion coefficient of the nutrient, 

 D n , and the radius, R, of the cell itself. Clearly the cell can do nothing 

 to control the value of C , the concentration of nutrient in the en- 

 vironment. It is by virtue of the dependence of <*>„ on C\ that the 

 cell is at the mercy of its environment. If C falls so low that $ s drops 

 below the minimum value for survival the cell dies or, if possible, 

 becomes dormant. Faced with a decrease in C , the cell might in- 

 crease $ s by increasing its radius R. All other things remaining the 

 same, a cell can increase its capacity to collect food in direct propor- 

 tion to an increase in its radius. It is doubtful that a cell can increase 

 in size without at the same time increasing its basal metabolic rate. 

 An increase in size would not be of much help if the demand for 

 nutrient increases as rapidly as, or more rapidly than, the increase in 

 the supply. Nothing would be gained by getting bigger in such a 

 situation. There is, however, a way in which the radius can in effect 

 be increased and that is by local stirring. Obviously the organism 

 cannot stir the entire bath, but conceivably it could be equipped 

 with some kind of organ, flagella perhaps, which with a small ex- 

 penditure of energy could stir the fluid in the immediate vicinity of 

 the cell. Such stirring would convey all nutrient molecules that en- 

 tered the stirred region to the surface of the cell where they would 

 be metabolized. The effect of local stirring would be to increase the 



