494 S. E. LURIA AND M. DELBRtlCK 



In contrast to this increase in the proportion of resistants on the mutation 

 hypothesis, a constant proportion of resistants may be expected on the hy- 

 pothesis of acquired hereditary immunity, as long as the physiological condi- 

 tions of the culture do not change. To test this point, accurate determinations of 

 the proportion of resistant bacteria in a growing culture and in successive sub- 

 cultures are required. In the attempt to determine accurately the proportion 

 of resistant bacteria, great variations of the proportions were found, and results 

 did not seem to be reproducible from day to day. 



Eventually, it was realized that these fluctuations were not due to any un- 

 controlled conditions of our experiments, but that, on the contrary, large 

 fluctuations are a necessary consequence of the mutation hypothesis and that 

 the quantitative study of the fluctuations may serve to test the hypothesis. 



The present paper will be concerned with the theoretical analysis of the 

 probability distribution of the number of resistant bacteria to be expected on 

 either hypothesis and with experiments from which this distribution may be 

 inferred. 



While the theory is here applied to a very special case, it will be apparent 

 that the problem is a general one, encountered in any case of mutation in uni- 

 parental populations. It is the belief of the authors that the quantitative study 

 of bacterial variation, which until now has made such little progress, has been 

 hampered by the apparent lack of reproducibility of results, which, as we shall 

 show, lies in the very nature of the problem and is an essential element for its 

 analysis. It is our hope that this study may encourage the resumption of quan- 

 titative work on other problems of bacterial variation. 



The aim of the theory is the analysis of the probability distributions of the 

 number of resistant bacteria to be expected on the hypothesis of acquired 

 immunity and on the hypothesis of mutation. 



The basic assumption of the hypothesis of acquired hereditary immunity 

 is the assumption of a fixed small chance for each bacterium to survive an at- 

 tack by the virus. In this case we may therefore expect a binomial distribution 

 of the number of resistant bacteria, or, in cases where the chance of survival 

 is small, a Poisson distribution. 



The basic assumption of the mutation hypothesis is the assumption of a 

 fixed small chance per time unit for each bacterium to undergo a mutation to 

 resistance. The assumption of a fixed chance per time unit is reasonable only 

 for bacteria in an identical state. Actually the chance may vary in some manner 

 during the life cycle of each bacterium and may also vary when the physio- 

 logical conditions of the culture vary, particularly when growth slows down on 

 account of crowding of the culture. With regard to the first of these variations, 

 the assumed chance represents the average chance per time unit, averaged 

 over the life cycle of a bacterium. With regard to the second variation, it 

 seems reasonable to assume that the chance is proportional to the growth rate 

 of the bacteria. We will then obtain the same results as on the simple assump- 



