142 FRANCIS D. CARLSON 



By assuming in the first approximation that this increased flux adds 

 directly to the flux due to diffusion, the steady-state flux for the mov- 

 ing cell becomes 



*. = 4 • x • RC ■ (d c + ?^\ 



If the cell executes not a linear but a random motion which can 

 be characterized by a "diffusion coefficient," D c , then, as was shown 

 by Chandrasekhar (1943) the flux relation becomes 



*c.,z>, + ^(. + * A)t]1( , ) 



where D c is the diffusion coefficient of the cell's motion. For t — > 

 oo, $ becomes 



«1> S = 4 • x • rC (D c + D n ) 



The diffusion coefficient of the cell is defined in terms of its motion 

 as 



6r 



where (r 2 ) is the mean square displacement in the time r. (r 2 ) is de- 

 termined by the statistical properties of the motion of the cell. D, 

 can be related to what might be considered an average velocity of 

 the particle by noting that 



<r) = (fr (r 2 r _ „ <r 2 r 



6r t 6 6 



where v = (r 2 ) 1/2 /r is the average velocity of the cell. This leads to 



<E> S = 4tt • RC (r> n + v- ^-j 



for the flux of nutrient. 



In summarizing the situation for the moving cell it can be stated 

 that the flux of nutrient across the boundary will in the first approxi- 

 mation depend linearly upon the velocity v, that is, <£, = 4tt • RC\- 



