456 HOWARD B. NEWCOMBE 



value obtained by means of method 2. This is important, inasmuch as there has 

 been no estimate of the magnitude of the bias in method 2, or of the extent to 

 which the discrepancy between methods 1 and 2 is a product of it. 



The above experiments demonstrate that this discrepancy is due to biologi- 

 cal rather than to statistical causes. 



There are two reasons for measuring mutation rate over periods of 6 and 

 of 11 generations. First, had it been measured over the shorter period only, it 

 would be possible for the results to be biased by a high mutation rate during 

 the first generation or so. The obtaining of similar estimates of rate from both 

 a short and a long period, however, eliminates this as an appreciable source 

 of error. 



Second, formula (3) assumes that the mutants multiply at the same rate 

 as the parent strain. That this assumption is approximately true is indicated 

 by certain experiments of of Luria and Delbruck and later of Demerec and 

 Fano. Somewhat more critical evidence however is obtainable from experi- 

 ments such as the above in which close agreement between estimates of muta- 

 tion rate from short and from long periods of growth indicate that there is no 

 appreciable bias arising from differential increase during logarithmic growth. 

 A mathematical demonstration of this will be found in a paper by Shapiro 

 (1946). 



For present purposes it is sufficient to state the argument in general terms. 

 From formula (3) it is evident that mutation acting alone causes the propor- 

 tion of mutants r/N to rise arithmetically with each successive generation. 

 If, however, the mutants were to increase more (or less) rapidly than the 

 parent strain there would be superimposed upon this an exponential increase 

 (or decrease) in r/N, and estimates of rate obtained using formula (3) would 

 tend to rise (or fall) correspondingly with increasing periods of exponential 

 growth. Thus it is clear that close agreement between estimates of mutation 

 rate over periods of 6 and 11 generations constitutes evidence that the high 

 estimates obtained using methods 2 and 3 are not the result of a differential 

 favoring the mutant during periods of logarithmic growth. 



It would be quite possible to supplement the above evidence on the relative 

 rates of increase of mutant and non-mutant strains, by preparing mixed 

 cultures and determining the change in proportion which takes place as the 

 result of competitive growth, as in the above mentioned experiments of Luria 

 and Delbruck, and Demerec and Fano. However the results of such experi- 

 ments are not entirely critical in the case of the spontaneously occurring mu- 

 tants, since the mutant strain must first be selected by growing in the presence 

 of phage, and then must be freed from all phage particles by suspending a 

 resistant colony in liquid, streaking the suspension on agar, and incubating 

 until visible colonies are formed, the process being repeated a number of times. 

 Such prolonged growth offers considerable opportunity for further mutation 

 and selection, and it is impossible to be certain that the strain which is finally 

 obtained will not have changed with regard to its ability to compete with the 

 non-mutant. 



This difficulty may be avoided, however, by the use of radiation induced 



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