GENERAL CHARACTERISTICS OF GROWTH-PROCESS 479 



bacteria is limited, and the rate of multiplication is slowed. McKen- 

 drick infers, therefore, that the rate of multiplication is proportional to 

 two factors; in the first place to the number of bacteria previously 

 produced, and in the second to the concentration of the still-available 

 foodstuffs. This leads to the equation: 



dx 



= kx(a x) 

 at 



where "x" is the number of bacteria per unit-volume, "a x" is pro- 

 portional to the concentration of available nutrients and "k" is a 

 constant proportionality-factor. Integration of this differential equa- 

 tion leads to the relationship 



log = ka(t - ti) 



a x 



where "x" is the number of bacteria per unit volume, a is the maximal 

 density of population which is attainable in a given culture medium, 

 "k" is a constant proportionality-factor and ti is the time at which 

 the density of the bacterial population has attained half its maximum. 1 



The relationship between the number or mass of bacteria produced 

 and the time of incubation which is expressed in these equations is, 

 however, identical with that which expresses the relationship between 

 weight and age in any given growth-cycle of an animal or plant. It is 

 also identical with the relationship between the mass of the products 

 and the time in autocatalyzed chemical reactions, such as the hydrol- 

 ysis of Methyl Acetate. The question therefore presents itself, whether 

 the process of growth in a multicellular organism such as a mammal is 

 comparable to an autocatalyzed chemical reaction, or whether McKen- 

 drick's interpretation of the growth-curve of a bacterial population does 

 not offer an alternative explanation of the facts. In other words two 

 alternative possibilities would appear to exist : the one that the accelera- 

 tive factor in growth is a chemical substance, as it is in autocatalyzed 

 chemical reactions, the other that it is simply due to the multiplication 

 of cells, each of which is possessed of like potentialities of reproduction. 



On closer analysis it will be seen, however, that these interpretations, 

 at first sight alternative, are in reality identical. 



Reverting to the case afforded by the multiplication of bacteria in a 

 limited amount of culture-medium, and looking to the beginning and 

 end of the process, we see that the increase in bacterial population 

 means essentially that the simple, unorganized constitutents of the 

 culture-medium have been transformed into the substances composing 

 the bacteria. Any acceleration experienced by the process must 

 ultimately be due to the preceding synthesis, irrespective of the fact 

 that the synthesis takes place in a heterogeneous system, i. e., in the 

 separate particulate masses which form the individual bacteria. When 



1 I have slightly, but unessentially, modified McKendrick's formulation of this 

 relationship in order to make clearer the analogies which follow. 



