MUTATIONS OF BACTERIA 495 



tion of a fixed chance per time unit, if we agree to measure time in units of divi- 

 sion cycles of the bacteria, or any proportional unit. 



We shall choose as time unit the average division time of the bacteria, di- 

 vided by In 2, so that the number N t of bacteria in a growing culture as func- 

 tion of time t follows the equations 



(i) dNt/dt = N t , and N t = N e 4 . 



We may then define the chance of mutation for each bacterium during the 

 time element dt as 



(2) adt, 



so that a is the chance of mutation per bacterium per time unit, or the "muta- 

 tion rate." 



If a bacterium is capable of different mutations, each of which results in 

 resistance, the mutation rate here considered will be the sum of the mutation 

 rates associated with each of the different mutations. 



The number dm of mutations which occur in a growing culture during a 

 time interval dt is then equal to this chance (2) multiplied by the number 

 of bacteria, 4 or 



(3) dm = adtNt, 



and from this equation the number m of mutations which occur during any 

 finite time interval may be found by integration to be 



(4) m = a(N t - No) 



or, in words, to be equal to the chance of mutation per bacterium per time unit 

 multiplied by the increase in the number of bacteria. 



The bacteria which mutate during any time element dt form a random 

 sample of the bacteria present at that time. For small mutation rates, their 

 number will therefore be distributed according to Poisson's law. Since the 

 mutations occuring in different time intervals are quite independent from each 

 other, the distribution of all mutations will also be according to Poisson's law. 



This prediction cannot be verified directly, because what we observe, when 

 we count the number of resistant bacteria in a culture, is not the number of 

 mutations which have occurred, but the number of resistant bacteria which 

 have arisen by multiplication of those which mutated, the amount of multipli- 

 cation depending on how far back the mutation occurred. 



If, however, the premise of the mutation hypothesis can be proved by other 

 means, the prediction of a Poisson distribution of the number of mutations 



* We assume that the number of resistant bacteria is at all times small in comparison with the 

 total number of bacteria. If this condition is not fulfilled, the total number of bacteria in this 

 equation has to be replaced by the number of sensitive bacteria. The subsequent theoretical de- 

 velopments will then become a little more complicated. For the case studied in the experimental 

 part of this paper the condition is fulfilled. 



