GENERAL CHARACTERISTICS OF GROWTH-PROCESS 481 



determining the magnitude of the inhibition. Either of these alter- 

 natives would yield the time-relations expressed in the autocatalyzed 

 reaction-formula, for the following reasons : 



In case the velocity of the reverse reaction is, at all stages of the 

 transformation, negligible in comparison with that of the forward reac- 

 tion, then the only inhibitive factor must be the exhaustion of the 

 Substrate, or material undergoing transformation. The velocity of the 

 process will be, as usual in chemical reactions, proportional to the mass 

 of untransformed material and also to the mass of the catalyzer, that is, 

 in these instances, to the mass of the products of the reactions. Desig- 

 nating the mass of a product of the reaction at any moment by "x," 

 and "a" the initial amount of the material undergoing transformation, 

 this yields the relation : 



dx 



Velocity of transformation = = kx(a x) 



dt 



which is the formula characteristic of an autocatalyzed reaction. 



Coming, now, to the case in which the velocity of the reverse reaction 

 is so considerable as to be comparable with that of the forward reaction, 

 we will assume, in the first instance, that the materials undergoing 

 transformation (or foodstuffs in growth) are inexhaustible, i. e., are 

 constantly being renewed from the environment, so that the mass of 

 material undergoing transformation is a constant which we may 

 designate by the symbol of "A." The velocity of the forward reaction 

 will then be, as in the above instance, proportionate to the mass of the 

 catalyzer ( = product of the reaction, = "x") and also to the constant 

 mass of substrate, that is, to "A." The velocity of the reverse reaction 

 (breaking-down of the products of the reaction into the initial sub- 

 stances again) will be proportional to the mass of the products ( = "x"), 

 but also to the mass of the catalyzer ( = "x"), because in the majority 

 of instances of "typical" catalysis the catalyzer accelerates both the 

 forward and the reverse reactions in equal proportion. The velocity 

 of the reverse reaction at any moment will therefore be proportionate 

 to x 2 , and the net velocity of the process, being the difference between 

 the velocities of the forward and the reverse reactions, will be given by : 



dx 

 - = kl *A = UP 



in which "ki" and "k 2 " are the velocity-proportionality factors of the 

 forward and reverse reactions respectively. Rearranging the terms of 

 the equation this may be written: 



-^ k 

 dt 



which is again identical with the ordinary formula of autocatalysis, 

 with the exception that the constant "a," denoting the maximal attain- 

 able value of "x" is now not the initial mass of material undergoing 

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