266 Distribution of numbers of mutants in bacterial populations 



inefficient statistics for estimating the parameters of the distribution from experimental 

 results, or for testing the agreement of experiment and theory. 



The purpose of the present paper is to extend Luria and Delbruck's method by calculating 

 the form of the distribution of numbers of mutants in parallel cultures to be expected on 

 the spontaneous mutation theory, so making the test of the applicability of the spontaneous 

 mutation theory a quantitative test. Statistically efficient methods of deducing the 

 mutation rate from experimental observations are also discussed. 



The distribution 

 First method 

 During the active growth of a culture, the number of organisms increases as an exponential 

 function of the time, and may be represented as 



n = eP*, (1) 



there being one organism at time t = 0. 



Thus dn = pndt. (2) 



If a is the mutation rate, defined by the relation that a dt is the probability that an 

 individual phage-sensitive organism shall undergo mutation to phage resistance in time dt, 

 n<x.dt is the mean number of mutations which occur in time dt. (Strictly, since n is the total 

 number of organisms, we should subtract from n in this formula the number of resistant 

 organisms, but in practice the number of mutant organisms in a culture is a minute 

 fraction of the total number.) Hence the mean number (m) of mutations which will have 

 occurred in the culture by the time it has grown to size n at time t is 



noidt = - 5 \ dn = -(n-l). 

 Jo pji R 



Since at all relevant times n much exceeds unity, we may write 



m = -n (3) 



for the mean number of mutations in the culture by the time it has attained size n. The 

 mean number of mutations which occur while the culture grows from n x to n 2 organisms 

 is evidently 



(4) 



P?) 



If it should happen that the mutation rate a and the growth rate fi are equally affected 

 by factors such as nutritional conditions and density of population which affect j3, then 

 the mutation rate per generation, though not per unit time, will be independent of these 

 factors, and a//3, which may be regarded as the probability of mutation per division, will 

 be constant even though a and /S are not. Under these conditions equations (3) and (4) 

 can be derived without the assumption of exponential growth. 



We must distinguish between the number of mutations which have occurred, and the 

 number of mutants, the latter being derived not only by mutation of the normal bacteria 

 but also by division of bacteria which have suffered mutation earlier. If r (r ^ m) is the 

 number of mutants in the culture at a given time, and if we assume that the division rate of 

 the mutant is the same as that of the normal bacteria, then rfidt = rdn/n is the probability 



26 



